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Theory of 
Mechanical Vibration 



Theory of 
Mechanical Vibration 

KIN N. TONG Professor of Mechanical Engineering 
Syracuse University 


Copyright © I960 by John Wiley & Sons, Inc. 

All Rights Reserved. This book or any part 
thereof must not be reproduced in any form 
without the written permission of the publisher. 

Library of Congress Catalog Card Number: 60-6460 
Printed in the United States of America 




To the memory of my father 



This book is the outgrowth of lecture notes for a course given to 
beginning graduate students and qualified seniors. Because of this 
origin, it is primarily a textbook, although some utility as a reference 
volume is also intended. 

A course in mechanical vibrations can be organized in one of two 
ways, which may be described as problem -centered and theory-centered. 
This book is written for a theory-centered course, which develops the 
basic principles in a logical order, with engineering applications inserted 
as illustrations. No attempt is thus made to cover all problems of 
technological importance or to restrict the discussion only to topics 
having immediate applications. It is felt that a theory-centered course 
has its place in an engineering mechanics curriculum, since the ana- 
lytical aspects of the theory have pedagogical values beside their 
utility in solving vibration problems. 

The book is divided into four chapters. Chapter 1 treats systems 
having a single degree of freedom. All the basic concepts pertaining 
to mechanical vibrations are presented, with the exception of vibration 
modes. Chapter 2 introduces the concept of vibration modes in a 
multidegree-freedom system, using a system with two degrees of free- 
dom as a simple model. The discussion is kept as close as possible to 
physical aspects of the problem. By means of matrix algebra and 
generalized coordinates, Chapter 3 extends the results previously ob- 
tained. In this way this chapter also lays the foundation for the solu- 
tion of vibration problems on digital computers and provides a heuris- 
tic picture of what is to follow. Chapter 4 discusses the vibration of con- 
tinuous media. Because only a limited amount of student knowledge 
in elasticity can be assumed, the systems selected for illustration in this 
chapter are relatively simple, yet the theory presented is quite general. 



The layout of this book is somewhat different from the usual. In the 
beginning of each chapter fundamental principles are presented in a 
connected series of articles. Articles dealing with examples, applica- 
tions, and specialized topics, which are more or less independent of one 
another, are placed at the ends of the chapters. (In the first three 
chapters these articles are grouped into two sections, A and B. The 
same grouping is not indicated in the fourth chapter, since there the 
demarcation is not so clear.) The purpose of this arrangement is two- 
fold. It emphasizes the structural coherence of the theory, and it 
affords flexibility in classroom assignments. The instructor can plan 
his lectures by following the main development of the theory. At 
intervals appropriate to the level and the interest of a particular class, 
he may discuss, or assign as home reading, examples, applications, and 
methods selected from this book or from other sources. A number of 
exercises is given at the end of each chapter. Many of' these exercises 
supplement the material in the text. 

The students are assumed to have the usual preparation, including a 
course in differential equations, in undergraduate mechanics and mathe- 
matics. Certain fundamental theorems in advanced calculus and in 
vector analysis are referred to in a few isolated passages; these can be 
omitted, if necessary, without disrupting the continuity of presenta- 
tion. An appendix on the basic ideas of matrix algebra is given. The 
scope of this appendix is limited, but it contains all that is needed for 
studying Chapter 3. In short, little prior knowledge is required to 
understand this book, although some degree of maturity is indispens- 

To keep the scope of the book within the limits of a two-semester 
course and to preserve the unity of the entire presentation, certain 
topics are omitted. These include nonlinear vibrations and the solution 
of transient problems by operational calculus. However, seeding ideas 
pertaining to these topics are planted in Arts. 1.4, 1.10, 1.11, 1.13, 1.14, 
2.8 and 3.8, but their complete development is left to other standard 
courses generally available to advanced students. 

Many persons helped to prepare this book. I wish especially to 
thank Professor Harold Lurie for a thorough reading of the manuscript 
and for offering valuable suggestions. Thanks are due to Mrs. Patricia 
Fisch and Mrs. Marilyn Levine for typing the manuscript and to Mr. 
C. Y. Chia and Mr. K. Ruei for assisting in various other ways. 

Kin N. Tong 

Syracuse, New York 
October 1959 


Introductory Remarks 


of Freedom 

Systems with a Single Degree 

Section A Theory and Principles 








Section B 


Introduction 3 

Simple harmonic motion 3 

Complex number and graphical representation of 

a sinusoidal function 6 

Harmonic oscillation of system with a single 

degree of freedom — General discussion 9 

Energy relation, Rayleigh's principle, and 

phase trajectory 10 

Damped vibration with viscous or linear 

damping 12 

Forced vibration under a harmonic force 17 

Complex number representation 24 

Steady-state response to periodic forces 28 

Work done by external forces and energy 

dissipation in vibratory systems 30 

Response of linear systems to a general external 

force — Superposition theorem 31 

Signal-response relation of linear systems in 

general 39 

Methods and Applications 

Examples of linear vibratory systems with a 

single degree of freedom 52 



1.13 Linearization of systems in small oscillations 55 

1.14 Piecewise-linear systems 59 

1.15 Theory of galvanometer and moving-coil 
instruments 68 

1.16 Seismic instruments and transducers 73 

1.17 Vehicle suspension 79 

1.18 Structural damping and the concept of complex 
stiffness 82 

Exercises 89 

of Freedom 

Systems with Two Degrees 

Section A 



Section B 


Theory and Principles 100 

Introduction 100 

Free undamped vibration — a model and its 

equation of motion 100 

Principal or normal modes 101 

General Solution 102 

Formulation by energy consideration — a 

generalized analysis for the free vibration of 

system with two degrees of freedom 105 

The use of influence coefficients 108 

Rayleigh's quotient 111 

Vibration of damped systems 114 

Forced vibration 116 

Degenerated cases 120 

Repeated roots in frequency equations — 

transverse vibration of rotating shafts 123 

Methods and Applications 128 

Illustrative examples 128 
Application of Rayleigh's method 132 
Some principles in vibration control 135 
Effects of rotation on critical speeds of shafts 146 
Exercises 161 

CHAPTER 3 Systems with Several Degrees 
of Freedom 

Section A 

Theory and Principles 
Introduction 168 



3.1 Generalized coordinates, constraints, and degrees 
of freedom 169 

3.2 Energy expressions in generalized coordinates for 
linear systems 170 

3.3 Summation convention and matrix notation 173 

3.4 Free vibrations of an undamped system — an 
eigenvalue problem 175 

3.5 Principal coordinates and orthogonal property of 
modal vectors 180 

3.6 Rayleigh's quotient 183 

3.7 Forced vibration of an undamped system 185 

3.8 Free and forced vibrations of a damped 
system 188 

3.9 Semidefinite systems 192 

3.10 Repeated roots of the frequency equation 196 

Section B Methods and Applications 197 

3.11 Solution of eigenvalue problems by matrix 
iteration 197 

3.12 Additional theorems and methods 208 

3.13 Chain systems — Holzer's method 212 

3.14 Electrical analog of mechanical systems and 
electromechanical systems 221 

Exercises 231 

CHAPTER 4 Vibration of Elastic Bodies 

4.0 Introduction 236 

4.1 Coordinates and constraints 237 

4.2 Formulation of a problem by differential 
equation 239 

4.3 Separation of time variable from space variables — 
reduction to eigenvalue problems 249 

4.4 Orthogonal property of eigenfunctions 260 

4.5 Formulation by integral equation 271 

4.6 Rayleigh's quotient and its stationary values 274 

4.7 Rayleigh-Ritz method 279 

4.8 Formulation of problem by infinite-series expan- 
sions of energy expressions — Rayleigh-Ritz 
method re-examined 287 

4.9 Forced vibration of elastic bodies 296 

4.10 Vibration of an infinite or semi-infinite elastic 
body — wave phenomenon 304 


4.11 Methods of finite differences 312 

Exercises 318 

APPENDIX. Outline of Matrix Algebra in 

Linear Transformation of Vectors 323 

INDEX 343 

Introductory Remarks 

Vibratory motions in machines and structures are of frequent concern 
in engineering practice. Such motion is usually objectionable; it may 
sometimes be desirable; and frequently its presence reveals to us the 
inner workings of complex machinery in operation. Whether our object 
is to minimize vibrations, to enhance them, or to utilize them for "diag- 
nostic" purposes, it is essential that the physical laws which govern such 
motions be studied. 

Vibratory motions are essentially periodic in character. Theory of 
vibration is therefore a special topic in analytical dynamics dealing 
principally with periodic motions of mechanical systems. In recent times, 
however, we have come to appreciate that the susceptibility of a mechani- 
cal system to vibrations describes certain intrinsic properties of the system, 
from which its general dynamic behavior can often be deduced even when 
periodic motion is not involved. Thus in a subtle way the principles 
governing the vibratory behavior of a system may often be advantageously 
used to supplement or even to replace the differential equations of motions 
in analyzing a number of problems in dynamics. 

Theory of vibration also claims kinship in varying degrees of removedness 
with such studies as acoustics, alternating current circuits, and electro- 
magnetic waves. All these studies concern themselves in part with periodi- 
cally changing phenomena which are governed by more or less unified 
principles; and the analytical results obtained in these disciplines often 
have interchangeable applicabilities. Furthermore, if we examine some of 
the analytical tools used in theory of vibration, we will also find that they 
resemble those used in a still larger class of physical and mathematical 
problems. These include problems in elastic instability, periodic structures 
and linear transformations. 

For the various reasons stated it then becomes justified to lift the study 
of vibratory motions from its parent body, analytical dynamics, and to 



place it on a pedestal of its own. In developing this subject, we have 
therefore not confined our attention strictly to the utility of the theory 
in solving vibration problems but have cast occasional glances upon many 
of its neighboring domains, into which we are likely to excurse in our 
future studies. Consequently, certain terms and concepts are introduced, 
not because of any immediate bearing they have upon the subject on hand 
but because of the desirability of acquainting ourselves with the many 
bridges that connect seemingly unrelated fields. 


Systems with a Single 
Degree of Freedom 


1.0 Introduction 

Our study begins with the discussion of vibrations of a system having 
only a single degree of freedom. We shall, however, refrain from defining 
at this moment precisely what is meant by such terms as "vibrations," 
"systems," and "degree of freedom." Instead, we shall try to build up our 
theory by analyzing some simple and readily visualized physical systems 
and leave the precise definitions of these terms to more opportune moments. 
In the meantime, terminologies not specifically defined are to be inter- 
preted according to their accepted meanings in mechanics, in mathematics, 
and in the English language. 

1.1 Simple Harmonic Motion 

The simplest form of vibratory motion is a simple harmonic motion, 
which is defined in kinematics as a rectilinear motion of a point whose 
acceleration at any time t is proportional and opposite to its displacement 
x. The mathematical description of this motion is therefore the differential 


W* = -° JX 

or 1 x + w 2 x = (1) 

1 The symbols x and x stand for the first and second time derivatives of the function 
x(t), respectively. This notation, employed originally by Newton, is used throughout 
this book, except when ambiguity may arise. 



in which co is a real number, hence the constant of proportionality, -co 2 , 
is always negative. A simple example of a motion described by (1) is that 
of the mass particle in the system shown in Fig. 1. This system consists of a 
linear coil spring having a spring constant k and a mass particle of mass 
m, which is attached to the spring and constrained to move along the axis 
of the spring. Let be the position occupied by m when all the external 
forces acting on m are in equilibrium, and let x be the displacement of m from 
at time t. If the displacement x causes a change only in the force exerted 
by the spring while all the other forces on m remain constant, 2 then according 






Figure 1 

to Hooke's law of elasticity and Newton's law of motion the acceleration 
of m due to its departure from its equilibrium position is given by 

mx = — kx 

mx + kx = (2) 

If we let 

«>*=- (3) 


we reduce (2) into the form of (1). 

The solution of the differential equation (1), that is, the relationship of x 
versus t which obeys (1), is known to be expressible in the general form 

x(t) = A cos cot + B sin cot (4) 

2 According to the way we formulate this problem, the question whether or not there 
is a gravity force is irrelevant. 


This expression has two constants of integration, A and B, which are not 
contained in the original differential equation and are therefore not 
determined by it. In other words, the physical constants of the system 
alone do not specify the motion uniquely without some other information 
concerning the actual motion itself. This information may be furnished 
by the initial condition, 3 that is, the displacement x and the velocity x 
at the instant t = 0. Substituting this condition in (4), and in the equation 
obtained by differentiating (4), we have 

x = A x = Bco (5) 

It is sometimes more convenient to express the solution in the form 

x{t) = C cos (cot — a) (6) 


C cos a = A C sin a = B 

C 2 = A 2 + B 2 a = tan- 1 - (7) 


In a simple harmonic motion the displacement is therefore a sinusoidal 
function of time, and the motion is a periodic oscillation characterized 
by the quantities co, C, and a. The quantity co, which is determined by the 
properties of the system, is called the circular frequency. It is related to the 
frequency f and the period T by 

co = 2nf T=- = — (8) 4 

/ 0) 

The quantity C, which is the maximum displacement in the oscillatory 
motion, is called the amplitude. Ordinarily, frequency and amplitude taken 
together are enough to give a complete description of a harmonic oscilla- 
tion, just as pitch and intensity are enough to describe a pure musical tone. 
The need here for a third quantity a, which is called the time-phase angle, 
therefore deserves further explanation. We see from (6) that a is an angle 
whose cosine gives the ratio of the displacement at t = to the amplitude 
of vibration. Hence the value of a depends on the designation of an 
instant corresponding to t = 0. For a truly periodic phenomenon, which, 

3 Although the term "initial condition" has the connotation of the condition at the 
beginning of the motion, the condition at any specific instant can be used for this 
purpose, and no generality is lost in setting / = for the instant in question. 

4 The difference between to and /is merely a matter of units. The conversion factor 
is the constant In radians per cycle. Hence, whenever this difference is immaterial tq 
our discussion, we use the term frequency for co also. 


as described by (1) and (6), has no beginning and no end, this designation 

is an arbitrary act, and the value of a has very little physical significance. 

If, however, the relations (1) and (6) come into effect only at the instant 

/ = 0, when a sudden change of external conditions affecting the system 

takes place, then the time-phase angle a contributes toward the description 

of this initial condition. For instance, if the spring-mass system is set into 

motion at t = by external means acting momentarily to give the mass a 

displacement x Q and velocity x , these quantities are then related to y. 

through the equation 

B x a 
tan a = - = — - (9) 

A x co 

When two periodic phenomena of the same frequency are studied 
together, the difference between their time-phase angles is called the phase 
difference. The phase difference measures how far these two periodic 
phenomena are out of step with each other. In many problems this phase 
difference rather than the absolute value of the time-phase angle has 
physical meaning. 

1.2 Complex Number and Graphical Representation of a Sinusoidal 

It is known in algebra that trigonometric functions are related to 
exponential function by Euler's formula: 

e i0 = cos + i sin 

For real values of 6 the real part of the complex number e* fl is cos 8. If 
we use the symbol "Re" to mean "the real part of." we may write (6) as 

x = C Re (V (w '- a) ) = Re fCe^'V" 1 ) 


= Re (Ae lMt ) 


/ = Ce~ H = C cos a — iC sin a = A — iB (11) 

This quantity / is called the complex amplitude of the sinusoidal function 

At this point it seems unnecessarily complicated and highly artificial to 
represent a cosine function by the real part of a complex exponential 
function. Certain advantages of this representation will, however, reveal 
themselves in time. 

A complex number 

a + ib = re id 



can be represented by a vector in the so-called Argand's diagram, as 
shown in Fig. 2. The length of the vector r is called the absolute value (or 
modulus) and the angle 6, the argument (or amplitude)^ of the complex 
number in question. Hence the absolute value of A is the amplitude of 
x(t), and the argument of A is the time-phase angle of x(t). In this way A 
gives a complete description of the integration constants, whereas the 
parameter of the original differential equation, viz., a>, is contained in the 
factor e iuit . In (10) we may say that e lwt represents the differential equation 
(or the system) and A represents a particular solution of the equation (or a 
particular motion of the system). 



(a + ib) 





Figure 2 

In Argand's diagram representation the complex amplitude A is shown 
as OL in Fig. 3. The complex number whose real part represents x at 
time / is represented by rotating OL through an angle cot to OX. The value 
of a: is then equal to the projection of OX on the real axis as it rotates around 
O with uniform angular velocity co. The vector OX is called a rotating 
vector. 6 When two or more rotating vectors with the same to are involved 
in a problem, it is only their relative positions that are usually of import- 
ance. Hence their positions may be "frozen" at a chosen instant, say 
/ = 0, when their relation is studied. For instance, if x(t) is a sinusoidal 
function having the frequency w and is represented by a rotating vector 
Xe iwt , then x and x are represented by rotating vectors 90 and 180° "ahead" 
of Ae ltot , respectively. The relative positions of these rotating vectors do 

5 It is unfortunate that the word "amplitude" is used both in the description of a 
complex number and in that of an oscillatory phenomenon. It becomes doubly unfor- 
tunate that the amplitude of one does not correspond to that of the other when a 
complex function is used to describe an oscillatory phenomenon. For this reason we 
use the word "amplitude" only in describing an oscillation and the word "argument" 
in describing a complex number. 

Those who are interested in the vagary of meaning of the word "amplitude" should 
also look up its definition in astronomy. 

6 It is also called a phasor or a sinor. 




not change as they rotate en masse with the same angular velocity ca. 
The position of a rotating vector at / = is, of course, the vector represent- 
ing the complex amplitude; therefore we often use the complex amplitude 
alone to represent a sinusoidal function of time if the frequency m is known 
to be constant and is of little consequence in the discussion. 

Figure 3 

It can easily be verified that the sum of two complex numbers is repre- 
sented in Argand's diagram by a vector that is the vectorial sum of the two 
vectors representing the two numbers. Furthermore, a little analysis (see 
Exercise 1.5) will show that superposition of two sinusoidal functions of 
time having the same frequency yields another sinusoidal function of the 
same frequency, whose complex amplitude is the sum of those for the 
original functions. Hence parallelogram theorem in Argand's diagram 
represents the rule for the superposition of sinuosidal functions of the 
same frequency. Rotating vectors of the same circular frequency therefore 


have the properties of vectors insofar as addition law is concerned. 
However, they do not possess all of the properties of vectors in the usual 

1.3 Harmonic Oscillation of Systems with a Single Degree of Freedom — 
General Discussion 

The discussion so far is essentially a study of the solution (and of its 
representations) of a differential equation. The spring-mass system shown 
in Fig. 1 is merely an illustration of a physical system whose motion 
is described by the differential equation. Needless to say, the results 
obtained in this study, being mathematical in nature, are applicable to the 
motions of all other systems in which the same mathematical formulation 
is valid. 7 We now discuss this class of systems in general. 

The spring-mass system illustrated is said to have a single degree of 
freedom because only a single variable is needed to specify the configura- 
tion of the system. By configuration we mean the location of all mass 
particles of the system in space. 8 A system generally consists of many, 
or infinitely many, mass particles. But in a system with a single degree of 
freedom the spatial coordinates of these mass particles are interrelated to 
one another and restricted to vary in such a way that only a single (but not 
unique) quantity is required to specify their locations. This restriction in 
their movements is supplied by what we call the constraints of the system. 
In the spring-mass system studied we have tacitly assumed that (a) the 
mass m is a rigid body, (b) external things, such as guides, restrict the 
motion of m to a translation, (c) the spring is weightless. 9 Under these 
assumptions, the configuration of this system at any instant can be repre- 
sented by a single function of time, x(t), called the coordinate of the system, 
for which we chose the displacement of m from its equilibrium position. 
Other choices can be made to serve our purpose equally well ; but, what- 
ever the particular choice may be, physical reasoning leads to the con- 
clusion that it must be a periodic function of time having the identical 

7 This observation may sound trite but upon it rests the most imposing edifice of 
mathematical physics. 

8 This means that in this definition we are concerned only with the geometrical state 
of the system, not with its general dynamic state, which involves also the velocities or the 
momentum of the mass particles, discussed in Art. 1.4. 

9 If you do not like the idea of weightless things, we can compromise the situation 
in the following manner. We assume that the mass of the spring is relatively small so 
that its deformation is largely produced by the forces acting at its two ends without any 
measurable contribution from the inertia force of the spring itself. In that case the 
displacement of each particle of the spring is determined by its total elongation and the 
laws of elastic deformation. (See also Art. 4.0.) 


frequency. In other words, the frequency of a vibratory system is a physical 
property independent of the choice of the coordinate. 

A number of vibratory systems with a single degree of freedom are 
illustrated in Arts. 1.12 and 1.13. However, for our present exposition we 
may continue to use the spring-mass system as a model without the loss of 

1.4 Energy Relation, Rayleigh's Principle, and Phase Trajectory 

The systems being studied are called conservative systems because they 
contain no mechanism for the dissipation of mechanical energy. Therefore, 
once such a system is set into vibration, it contains a fixed amount of 
mechanical energy. This amount is divided between the potential and the 
kinetic energy. The vibration of the system can be considered as a periodic 
transferring back and forth of the energy from one form to the other. 
At the instant the system assumes its equilibrium configuration, its 
potential energy is a minimum, 10 or zero, and its kinetic energy, a maximum. 
When the system is momentarily at rest, the reverse is true. The maximum 
potential energy and the maximum kinetic energy occurring at these two 
extreme configurations obviously must be equal. 

Using T and V as symbols of kinetic energy and potential energy, 
respectively, we have for a simple spring-mass system 

T = imx 2 V = \kx 2 

Since the motion is a harmonic oscillation and the maximum kinetic 
energy T m and the maximum potential energy V m are equal, we have 

T m = imco 2 C 2 

V m = \kC* 

T m = V m or to = Vkf^i (12) 

In many problems in which to is to be determined it is easier to express the 
energies in terms of the amplitude C than to write the equation of motion, 
especially when approximations are involved. This method of determining 
the natural frequency of a system in harmonic oscillation is called Rayleigh's 
method. Its application to vibration problems is illustrated in Art. 1.13 
The reader is advised to study these illustrative problems before pro- 
ceeding further. 

10 Recall the theorem in mechanics which states that the necessary condition for a 
system in stable equilibrium is that its potential energy is at a minimum. 


In a more general discussion both m and k in the energy expressions 
must be considered as functions of the coordinates x. Hence 11 

T = \m(x)x 2 V = \k(x)x 2 

For conservative systems the conservation of energy demands that 

T + V = constant 

\m{x)x 2 + \k(x)x 2 = \h 2 

h l \m h 2 \k 

in which h is a constant. 

Consider a coordinate system in which the coordinate axes are the 
displacement x and velocity x. A generic point P in this coordinate plane, 
which is called the phase plane of motion, represents a dynamic state of 
the system, specified by the displacement and velocity taken together. The 
motion of the physical system is represented by the motion of P in the 
phase plane. The locus traced by P is called the phase trajectory, and the 
velocity of P in the phase plane along the phase trajectory is the phase 
velocity. Much about the motion of a system can be revealed by studying 
the topological structure of its phase trajectory in the phase plane. 

The phase trajectories of a given system are described by (13), in which 
m and k are determined by the system and h is a parameter determined 
by the initial condition (or energy content) of the system. For conservative 
systems in oscillation the phase trajectory must be a closed curve. The 
totality of all possible oscillations for a given system is represented by a 
one-parametric family of such curves. 

According to (13), the phase trajectory of the motion of a spring-mass 
system in which m and k are constants is an ellipse, shown in Fig. 4. The 
ratio of the two axes of the ellipse is the circular frequency of the system, 
and their absolute lengths are determined by the total energy content of 
the system in motion. Therefore, one may say that the constants of the 
system determine the shape of the ellipse, and the initial conditions, the 
size. 12 For a given system, through each point in the phase plan except 

11 How x and x enter into these expressions is discussed more fully in Chapter 3 and 
is of no importance to our present discussion. See also Art. 1.13 and Exercise 1.26. 

12 It is an important property of a linear system that its natural frequency is independ- 
ent of its amplitude. The definition of the term "linear system" is discussed more 
thoroughly in Art. 1.10. 




the origin, there passes a unique phase trajectory (ellipse). By giving the 
system an initial condition or dynamic state corresponding to this point, 
the resulting motion is depicted by the ellipse passing through this point. 


1 . 






„ h 

Figure 4 

1.5 Damped Vibration with Viscous or Linear Damping 

A viscous damper is a device that offers a resistance to motion propor- 
tional to the time rate of its deformation. If such a device is incorporated 


Figure 5 

into the spring-mass system studied, as shown in Fig. 5, the resulting 
equation of motion is 

mx = —kx — ex 
or (14) 

mx + ex -f kx = 




The constant of proportionality c is called the damping constant, or simply 
damping. The general solution of (14) is given by 

x(t) = Q^ + C 2 e s ^ (15) 


— c ± Vc 2 — 4mk 


and Q and C 2 are constants of integration to be determined by the initial 
condition. There are three possible cases of the resulting motion. 

Figure 6 

Case 1. Over-damped systems 

c 2 — 4mA: > 

In this case s x and s 2 are both real and negative. For the motion with the 
initial condition 

x ^= Xq x ^ = Xq at t ^ = u 
the constants are 


^2^0 i ^0 

C 9 = 

SjXq + ^o 


°1 °2 °2 °1 

The resulting motion may be one of the types represented in Fig. 6. 
Depending upon the relative magnitudes of x and x , the curve may cross 
the /-axis at most once. (See Exercise 1.20.) At any rate, no oscillation 
may take place ; and x and x approach zero asymptotically. The motion is 
said to be aperiodic. 

Case 2. Critically damped system 
c 2 — 4mk = c 

= iVmk 


For this particular value of damping, c c , called the critical damping of the 
system, we have 

s i = s 2 — ~ 0) 

and the solution of (14) becomes 

x(t) = C 1 e~ (o1 + C 2 te- M (18) 


Cj = Xq C 2 = x ~r ojXq 

The motion does not differ from that of Case 1 in its essential character. 
It is also aperiodic. 

Case 3. Under-damped system 

c 2 - 4mk < 

In this case s x and s 2 are a pair of complex conjugate numbers with 
negative real parts. The solution is more conveniently expressed as 

x(t) = e { - cl2m)t (A cos co e t + B sin co c t) (19) 


x(t) = Ce { - Cl2m » cos (o) c t - a) (19a) 


V4mk — c 2 I ( c 2 

1--J (20) 

2m v \ c v 

and the relationships among A, B, C, and a are the same as in (7). The 
motion is therefore oscillatory in nature and is often called a damped 
oscillation. It can also be conveniently thought of as a sinusoidal motion 
with diminishing amplitude. 13 

In complex number representation (19) and (\9a) may also be written 

x = Re (Xe iat ) (21) 


X = Ce- ix = A - iB 

a = co c + i — = oi c + i— co (22) 

2m c c 

The quantities X and cr are called the complex amplitude and the complex 
frequency of a damped vibration. From (22) it is seen that a is a property 
of the system, whereas ?. describes the initial condition. 

13 To say that this description is mathematically unacceptable, since a sinusoidal 
motion must have a constant amplitude, is "justified quibbling." 




= x at 

/ = 

Xq = 

Re (A) = A 

Xq = 

Re {ioX) 


. C c X 
= x n - I 




X = A-iB = x -i ~"" u ' u (23) 

co c c c 

Returning to (19), we see that the ratio of two displacements at any two 
instants a period, T = 2rrj 'co c , apart is a constant. 

X {*' Jrl9m\T (1A\ 

x(t + T) 

In other words, at regular time intervals of 2ttJ(o c the displacement 
decreases in a geometric progression. The natural logarithm of this ratio 
is called the logarithmic decrement, denoted by the symbol A. 

x(t + T) 2m o) c 

The value of A is a measure of the amount of damping the system 
possesses. In many actual systems the damping force is not so localized 
that the damping constant c can be directly measured. If such a system 
is set into vibration, one can observe A and deduce an equivalent damping 
constant for the system by the following relationships: 

c 2tt c to 

A = = 2tt 

2m co c c c co c 

Electrical engineers, for good reasons, generally prefer to use a quantity 
called quality factor Q to describe the damping property of a circuit. The 
quantity Q can be defined by 

Q = - 
* A 

From (20) we also have 




Thus the three ratios ojJco, c/c c , and A/l-n- can be expressed as three 
trigonometric functions of a single angle d, as shown in Fig. 7. 

sin d = 

C c 


c oj c 
cos = — 



. A 
tan o = — 



Theoretically, the equivalent damping constant can also be obtained 
from measuring oj c , but practically this cannot be done with accuracy 
when the amount of damping is small. For example, it can be seen in 






w c 


Figure 7 

(26) that with a damping of 20 per cent of the critical value there is only 
less than 2 per cent difference between eo and co c . This also shows that the 
introduction of a moderate amount of damping changes the frequency of a 
vibratory system only slightly. Furthermore if the damping is small, 

l-***± (28) 

C c 277 

A graphical representation of the motion can now be constructed, as 
shown in Figs. 8 and 9. We see from (21) that the displacement x is 
represented by the projection on the real axis of the vector in Argand's 
diagram representing the complex number Xe [at . This vector rotates with 
angular velocity eo c and decreases in length exponentially. The end of the 
vector describes a logarithm spiral}* which has the geometric property 
that the angle between its radius vector and its normal is constant. Let 
it be an exercise (Exercise 1.12) to prove that this angle is the angle 6 
defined in (27). (Work Exercise 1.11 also.) 

14 A logarithm spiral has many other interesting geometric properties. It fascinated 
the famous mathematician Jakob Bernoulli (1654-1705) so much that he willed it to be 
inscribed on his tombstone. Unfortunately, the execution of his will was not done with 
mathematical care and the inscription appeared to resemble an Archimedian spiral. 




Figure 8 


Figure 9 

1.6 Forced Vibration under a Harmonic Force 

(a) undamped system 

Let an external harmonic force, that is, a force which is a sinusoidal 
function of time, be acting on the spring-mass system shown in Fig. 1. 
The differential equation of motion may be written 

mx + kx = F cos co f t 



in which F is the amplitude of the force and to f is the circular frequency. 15 
The solution of (29) is 


x = A cos cot + B sin cot + z — z cos a>J (30) 

k — moj f 2 

in which A and B are constants of integration to be determined by the 
initial condition. Let this be x = x and x = x at t = 0. Then 


^o B = — 

k — mco f 2 co 

A = x - - ^— 2 B = - u (31) 

x F F 

x = x cos cot + — sin art — — - cos cot +' — - cos co f t (32) 

co k — nuof k — mcvf 

Several interpretations for the physical meanings of the terms in (32) 
are possible. Let us consider the following one. Suppose prior to t = 
the system is in free vibration, and at t = 0, when its displacement and 
velocity are x and x , respectively, an external force F cos co f t begins to 
act on the system. Both the equation of motion (29) and its solution (32) 
are then valid for t > 0. Comparing (32) with (4) and (5), one sees that 
the first two terms of (32) represent the motion that was in existence at 
t = and has the natural frequency of the system co. The third term 
represents the motion that is produced by the sudden application of a 
force at t = and also has the natural frequency co. These two motions 
are therefore due to conditions existing and to things happening at t = 0. 
They are called the transients because they eventually die out if there is 
any dissipation mechanism in the system, however small. 16 On the other 
hand, the motion represented by the last term has the frequency of the 
external force and is ever present as long as the force continues to act on 
the system. This motion is called the steady-state response of the system 
to the force F cos co f t. In many practical problems one is merely interested 
in motions that persist. An analysis that ignores the transient terms is 
called a steady-state analysis. 

15 The physical meanings of the words "amplitude" and "frequency" when applied 
to a harmonic force should be obvious. 

16 The decay of these transient terms is analogous to that of (19) if there is any damping 
force. Physical reasoning also shows that unless a system is completely free from 
disturbances or is unstable its condition at time t = 0, which determines the magnitude 
of the first three terms in (32), must have decreasingly small influence on its condition 
at subsequent instants. Note also that the quantity F in (31) is better interpreted as the 
value of the force at t = than the amplitude of the harmonic force. 


The steady-state response of the system being analyzed is thus 


k — mco f 

2 C0S "V=, _, , tf 

1 - (m,la>y 

cos co f t = |A| cos 0) f t 



where d si = Fjk is called the static deflection, which is the deflection of 
the spring under a static force equal to the amplitude of the external 
harmonic force. 17 

The nondimensional ratio U|/<5 S/ , called the magnification factor, 18 
versus the frequency ratio co f jco is plotted in Fig. 10. It is seen that for 

Figure 10 

forcing frequency co f below the natural frequency co, \?.\ and F have the 
same sign, and the displacement is in phase with the force. As co f increases 
toward co, \k\jb st increases from unity to larger values. 

According to (33), for (o f larger than co, F and |A| have opposite signs. 
This is to be interpreted that the displacement is 180° out of phase with the 
force. The amplitude decreases with increasing «j f and approaches zero 
as a limit. At co f = co the amplitude is indeterminate, and a condition 
called resonance exists. 

To study what takes place at resonance, let us consider first the case in 
which (o f is nearly equal to but slightly smaller than co, or 


co f = 2 A co 

17 Note carefully the definition of d st . Although it may have other physical meanings 
in other special cases, the definition given here is a more general one. 

18 Further discussions on "magnification factor" are contained in this article and in 
Art. 1.11. 




Assuming in (32) that the system is at rest at t = 0, that is, x = i e = 0, 
we have 


k — mco f 2 

(cos CO f t — cos cot) 
(oj f + co)t 

1 - (cof/cof 
u 2 d st 

Ao) (GO + COf) 
COf + CO 

2 sin 

sin A a> t sin 

sin Acd t 

(co f + oj)t 

> Ao> 


within a few cycles of sin (co f + eo)//2, the function sin Aoj t changes only 
slightly. The resulting motion can be represented by a series of sine waves 

sin Aut 


Figure 11 

whose amplitudes change slowly within envelopes which are two sine 
waves of frequency 2Aco, 19 shown in Fig. 11. 

Returning to (34), we now let Aco— > 0; then 

sin Aco t— > Aco / 

lim x(i) = b st {cotj2) sin cot 



19 Since sin Aco t enters as a part of the amplitude only, its absolute value alone is 
important, and this value repeats itself every 180° instead of 360 \ 




Thus the amplitude grows indefinitely as / increases. 20 The result is 
shown in Fig. 12. This condition is called resonance. It is important to 
note that the motion is a sine function, whereas the force is a cosine 
function. Hence the displacement lags behind the force by a phase 
difference of 90° instead of or 180°, as in the case of nonresonant forced 

The motion described by (34) is an example of a phenomenon called 
beat. It is formed by two harmonic oscillations with the same amplitudes 


Figure 12 

but slightly different frequencies. The phase difference between the two 
changes continually so that sometimes the two oscillations reinforce each 
other and at other times they cancel each other. In a more general beat 
phenomenon the amplitudes of the two need not be the same. The general 
case is therefore represented by 

x(t) = A cos (co + Aco)/ + B cos (co — Aco)/ 

= (A + B) cos t cos cot — (A — B) sin cot sin Aco t 

= C cos (cot — a) (36) 


C 2 = A 2 + B 2 + 2AB cos (2 Aw /) 

tan a = - --- tan (Aco t) 

A + B 



20 This is a more precise statement than the one that asserts that the amplitude is 
infinite. Aside from physical limitations, such as overstraining and breaking of parts, 
it takes time to build up large amplitude and infinite time to build up infinite amplitude. 
True steady-state forced vibration of an undamped system is not possible at resonance. 

21 In electronics we call the variation of C with t amplitude modulation and that of a 
with t, frequency modulation. 




The foregoing relations are nothing but the law of cosine and the law of 
tangent applied to the triangle shown in Fig. 13. The maximum and 
minimum "displacements" are therefore approximately the sum and the 
difference of the amplitudes A and B, respectively. The beat frequency is 

of LAOB 

2 A art 

Figure 13 

Aoj/7r, or the difference between the original frequencies. In Fig. 14 the 
two different cases of combined motion are shown: (a) co = Aeu and (b) 
co ^> Aco. It is to be noted that in (b) the envelopes are not sine waves, 
being somewhat "pinched in" in the valleys, especially when A and B are 

co = Aco 


Figure 14 

about equal. When A and B are equal and a = 0, the bottoms of the 
valleys become cusps, and the two envelopes are joined there to form 
sinusoidal envelopes, as shown in Fig. 11. 

In forced vibration near resonant frequency, if the system has a slight 
amount of damping, the beat will die out, since one of its component 


vibrations is a transient. On the other hand, if the system is subject to 
slight disturbances from various sources at all times, beats of various 
amplitudes but the same beat frequency will be observed continually. The 
important point is that since the transients have a frequency very near to 
that of the steady-state vibration during their brief time of existence they 
Can make their influence distinctly observable by forming beats with the 
steady-state vibration. 

(b) damped system 

We consider next the steady-state response to a harmonic force of a 
system with viscous damping. Let the equation of motion be 

mx + ex + kx = F Q cos w f t (38) 

The solution of this equation is known to be of the form 22 

x(t) = C^ + C 2 e s * 1 + |A| cos (a> f t - a) (39) 

in which |A| and a are determined by substituting (39) into (38): 

— mco f 2 + k = (F /|A|) cos a 

cco f = (f /|A|) sin a 

coj f 2(cjc c )((x) f lcQ) 


tan a = 

—moo 2 + k 1 — (cOf/co) 


^o A. 

V(-mw f 2 + kf + c 2 co/ 

\ CO 2 J \ C r 0) ) 

These relationships can be geometrically represented by a parabola, as 
shown in Fig. 15. The validity of this representation, first given by C. 
Runge, can be readily verified. By either the analytical expressions of (41) 
or the graphical construction shown in Fig. 15, graphs of the magni- 
fication factor |/l|/(5,, and of the phase lag a as functions of co f jco and cjc c 
can be plotted as shown in Fig. 16. They represent the so-called frequency 
response of the system. A general discussion on the frequency response of 
linear system is contained in Art. 1.11. 

The maxima and minima of the curves in Fig. \6a are of practical 

22 Although (38) can be solved by standard methods for solving differential equations 
of this type, we employ the following intuitive reasoning. The solution consists of a 
transient that must be the same as the solution of (14) and a steady-state motion that 
must be of frequency o) f . The steady state, however, may or may not be in phase with 
the force, so we introduce a phase difference a. 




interest, as illustrated in Art. 1.15. For c/c c < 1/V2 all the curves possess 
a maximum lying between frequency ratios of and 1 and a minimum 
at (0, 1). The point (0, 1) becomes the maxima of all curves with damping 
ratio greater than 1 / V2. Exercise 1 . 1 7 is devoted to the verification of these 

The curves for the phase lag a, shown in Fig. 16b change gradually 
from to 77 as co f increases. For different values of c\c c the curves first 
diverge from the origin, converge at (1, tt/2), diverge again, and then 
approach tt asymptotically. 

5st ~-^ > 


■^^^ ^-Parabola 

1 \ 


Figure 15 

1.7 Complex Number Representation 

The amplitude of the steady-state response |/| in (39) is assumed to be a 
real number. By replacing it with a complex number, it can take into 
account the phase difference a ; and we may write 

x = Re (Xe^) = Re (f) 
F cos (a t t = Re (F e i(O ' t ) 


Since (42) satisfies (38) 

m -^ Re (0 + c i -Re (£) + k Re (£) = Re (F e^ 

Since ra, c, and fc are real constants and t is a real variable, 
Re {ml + c£ + *£) = Re (F e^) 



Now if we change our "zero point" on the time scale and make the external 
force F sin co f t instead of F cos co f t, it is evident that the steady-state 




S s t 



-c/c c 



-c/c c 

= ¥ 



-c/c c 

= _ 5" 

c/c c 

~ 2 "~ 





:= 1" 



1 2 



response of the system will not be altered in its physical character, although 
its analytical expression is changed accordingly. In other words, the 
motion is still of the same amplitude and frequency, and it maintains the 
same phase difference with the force as before. Thus its analytical ex- 
pression is the same as the last term in (39), except that the cosine function 
is changed into a sine function. Therefore the imaginary part of £ satisfies 

Im (ml + ct + kQ = Im (F e ico ^) 
By combining the two we obtain 

ml + cl + kZ = F e iuj f l 


-mco/Xe^ + icwjie*** + Jt^ = F Q e iw f l 


F F lk 

A = 

—ma) f 2 + k + icco f i co f\ 2 .(2 C( °f\ (46) 

\co 1 \C C CO J 

The absolute value of A, which we denote by the symbol |A| is then 



VC-may 2 + kf + cho f 2 

and the argument of A, which is the phase difference between the force and 

the displacement, is 


-a = arg (A) = -tan" 1 f— — 

— mo/ + k 


tan a = ^— (48) 

— mo)/ + k 

These are identical to the results previously obtained. 

The purpose of this analysis is to show that from the viewpoint of 
solving problems it is not really necessary in complex-number representa- 
tion to take the real part only. We do not have to make the differentiation 
between x and £. and it is just as well if we write (38) as 

mx + ex + kx = F e ia ^ (49) 

with x — Xe iu)ft as the steady-state response. This is permissible if, and 
only if, all the terms contained in equations such as (38) are sinusoidal 
functions of the same frequency; because then the rotating vectors, 
representing the various terms, maintain fixed positions relative to one 
another as they rotate with the same circular frequency. Any additive 
relationship existing among their projections must also hold vectorially 


among themselves. The vectors, which represent the inertia force mx, the 
damping force ex, and the spring force kx, are shown in Fig. 17. Their 
vectorial sum is the vector representing the external force. 23 

It cannot be overemphasized that in applying mathematical tools to 
solve a physical problem a set of "ground rules" is always agreed upon 
beforehand by the users. Literal interpretation of symbols beyond what 
the ground rules allow is meaningless. 24 In the present analysis it is 
understood that we are interested only in getting the amplitude and phase 
difference of the steady-state response of a linear system to an external 
force which is a sinusoidal function of time. Under such circumstances, it 
has been shown that with proper representation complex-number algorism 

Figure 17 

can yield the correct results. No claim is being made that all physical 
aspects of force and displacement can be adequately described by the 
mathematical properties of complex numbers. 

In the transient analysis we have shown that a damped oscillation can 
be represented by the real part of a complex number 

x(t) = Re (£) = Re (Xe iot ) 

It is rather simple to verify that the imaginary part of £ also satisfies (14). 
Hence the complex number itself also satisfies 

ml + ct + k£ = (50) 

and, with proper understanding, we may dispense with the difference 

23 The parallelogram law for the addition of coplanar vectors is equivalent to the 
addition rule of complex numbers. Using mathematical terminology, one says that 
coplanar vectors and complex numbers are isomorphic under addition laws. 

24 Even the common practice of representing a scalar physical quantity by a real 
number involves a prior agreement of units and method of measurement. It is meaning- 
less to say, for instance, that boiling water is 212/32 times as hot as ice. 


between x and £. Our problem becomes that of looking for a solution of 
(50) of the type £ = le iat , satisfying the initial condition specified by (23). 

y i • C c x ~T" COJX 

bo — A — X ~ l 

At this moment some of the keener readers may recall that according 
to the theory of differential equation a second-order equation requires 
two, not one, initial values for its complete solution whereas only one is 
given by (23). There are several ways to explain this discrepancy, but all 
are based on the fact that in our scheme of things the other solution of 
(50), viz., £ = X'e tat (where a is the complex conjugate of a), can be 
ignored, since, being a mirror image of the other in Argand's diagram, 
it represents nothing that is not already included. Because we have only 
one integration constant, K, to determine, we need only one initial value. 

1.8 Steady-State Response to Periodic Forces 

It is known that a periodic function of time can be represented by a 
Fourier series; and under a wide range of conditions, which are usually 
satisfied in a physical problem, the series converges to the value of the 
function almost everywhere. 25 


F(t) = ^ ( a n cos nco ft + b n sin noj f t) 


00 00 

= 2 /. cos (nco,t - pj = J Re (<f>„e<"^) 

»=0 »=0 

where T = lirjcOf is the period of the function and for each n the constants 
a n , b n , f n , f$„, and <j> n are interrelated in the usual manner. It is well 
known that the coefficients a n and b n are given by 

«0 = r\ F W dt ^0 = 

1 J-T/2 

2 f T/2 

a n = - F(t) cos nco f t dt (52) 

1 J-T/2 


2 C TI2 

b n = — F(t) sin nw f t dt 

T J-T/2 

25 See standard mathematics text about Dirichlet's conditions in Fourier-series 
representation. The term "almost everywhere" has precise mathematical meaning with 
which we need not be concerned here. Sufficient is it to say that the series will converge 
to the function at all points of continuity. 


A periodic force, when expressed by a convergent Fourier series, is thus 
the sum of sine and cosine functions. By the principle of superposition, 
which is discussed fully later, the response of a linear system to a periodic 
force can be expressed as an infinite series whose terms are the responses to 
the terms in the Fourier series, provided that both series are convergent. 
Utilizing the results previously obtained, such as (33), (39), and (41), we see 
that the steady-state response to a periodic force for an undamped spring- 
mass system is 

1 °° f 

x = t 2 : r-TT2 cos ("<V - Pn) (53) 

k n=o 1 — {ncOflcoY 

and for a damped spring-mass system, 

x = 2 y fn cos (nco f t - n - a B ) 

kn=o // n 2 co 2 \ 2 (2cncoA 2 (54) 

n 2 co f 2 \ 2 1 2c nco f \ 
co 2 ) \c r co / 


(2clc c )(nco f lco) 

^ = tan ~\ — 7 — r^ 

1 — {ncOfjcoY 

Both expressions are a Fourier series; the motion they represent is 
therefore periodic and has the same period as that of the force. In com- 
paring the coefficients of these series with those of (51), it is seen that unless 
co is an integral multiple of co f the convergence of (53) and (54) is assured. 
As a matter of fact, the series representing the motion converges much 
faster than that representing the force, and one need not use too many 
terms of (51) to obtain approximate answers for x in (53) and (54). 26 

The studying of a periodic phenomenon by resolving it into sinusoidal 
components is called harmonic analysis. Besides the formal mathematical 
operations expressed in (52), there is a mechanical or electronic device, 
called harmonic analyzer or wave analyzer, which will "measure" the 
coefficients/^ (and sometimes also /?„) of a periodic phenomenon. Numeri- 
cal procedure for computing the coefficients in the Fourier series, repre- 
senting a physical phenomenon based on a finite number of measurements 
taken on the phenomenon, is also available. 27 

26 Note also that a n -* -n as n -> co. 

27 For details see Scarborough, Numerical Analysis, Johns Hopkins Press, Baltimore, 
1955, 3rd edition, pp. 477-494. 


1.9 Work Done by External Forces 

and Energy Dissipation in Vibratory Systems 

The work done in a time period T by a force F(t) acting on a body 
moving with velocity x(t) in the direction of the force is given by the 

W= \ T F(t)x{t)dt (55) 


If both the force and the displacement are sinusoidal functions of time 

having the same frequency co f and a phase difference of a, then the work 

done per cycle of the motion is 


r2ir/(o f 

W = F cos co f t -^ |A| cos (co f t — a) dt 


= 7rF |A| sin a 

Thus it is seen that work done per cycle is not only proportional to the 
amplitudes of the force and the displacement but also to the sine of phase 
difference. In addition, there is the constant of proportionality 77. In 
steady-state forced vibration of an undamped system a is or 77, except 
at resonance. The work done per cycle is therefore zero, and no energy 
is accumulated. At resonance, as discussed in Art. 1.6, a becomes 77/2, 
and there is a net amount of work gained by the system for each cycle 
proportional to the amplitude of that cycle. Since the system contains no 
mechanism for energy dissipation, the amplitude must build up each cycle 
by an equal amount. 28 This accounts for the straight-line envelopes in 
Fig. 12. For the steady-state vibration of a damped system sin a does 
not vanish, the net work performed by F is dissipated by the damping 
mechanism in each cycle, and a constant amplitude is maintained. Since 
the damping force is the only force that can dissipate mechanical energy. 
the work done by the damping force per cycle must be equal to that by 
the external force. Hence 

ttF\X\ sin a = cx(t)x(t) dt 


V 2 sin 2 (co f t -<x)dt = ttc|A| 

= C H 


F c(D } 

|A| sin a 



28 The truth of this statement lies with the fact that the total energy of the system, E, 
is proportional to the square of the amplitude. E oc |A| 2 and (dE/dt) oc |/.|; hence, 
d\)\\dt = constant. 


By the use of (47) and (48) it can be readily verified that the foregoing 
relationship is correct. For systems with a small amount of damping |A| 
is almost maximum at a = 77-/2. (See Fig. 16 and Exercise 1.17.) The 
damping of the system can thus be measured by exciting the system to 
vibrate with maximum steady amplitude and compute c from the approxi- 
mate relationship 


1 dT 

1 dV 



K =: 

C :== — 

xx dt 

xx dt 

X 2 



The energy relationship also serves as a convenient means for setting 
up the differential equation of motion, especially when the system contains 
a number of constraint forces, which do no work and can thus be ignored 
in the analysis. An example is given in Art. 1.12. The basic energy equation 

d d dW 

dt dt dt 

in which P is the power of the dissipative force and W is the work done 
by external forces. The parameters of a single-degree-freedom system can 
then be defined by 

^ \dW 
and F = 

x dt 

Depending upon the coordinate variable x chosen, the quantities m, k, c, 
and F, may or may not have their usual dimensions. 

1.10 Response of Linear Systems to a 

General External Force — Superposition Theorem 

The systems discussed so far are called linear systems because their 
equations of motion are linear differential equations having the form 

L(x) = F(t) (60) 

where L is a linear differential operator. 

L = a ° (0 a¥ + a±(t) aW^'" an ~ l(t) dt + aM 

For the systems studied n = 2 and the a's are constants instead of 
functions of time t. The following discussion is, however, pertinent to 
general linear systems. 

An important property of linear systems is that they follow the principle 


of superposition. This property is derived from the fact that linear opera- 
tions are distributive ; that is, 

L(x 1 + x 2 ) = L(^) + l(x 2 ) 

Thus if x = x ± (t) is a solution of L(x) = F ± (t), satisfying the initial con- 
dition x(0) = x 10 and x(0) = x 10 , and x = x 2 (t) is a solution of L(x) = 
F 2 (t), satisfying the initial condition x(0) = x 20 and x(0) = i 20 , then 
x = ax-^t) + /fo 2 (0 is a solution of the differential equation 29 

l(x) = aF x (r) + /JF 2 (0 

and satisfies the initial condition 30 

x(0) = ax 10 + fix 20 x(0) = ai 10 + f3x 20 

in which a and /? are arbitrary constants. 

Let the equation of motion of a linear system under the influence of an 
external force F(t) be 

L(s) = F(t) (61) 

and let x = u(t) be the solution of the differential equation 

l(x) = S(t) (62) 

satisfying the initial condition x = 0, x = 0, in which J~(r) is a unit step 
function defined as 

J"(0 = for t < 

J"(0 =1 for t > 

This function «(/) is called the indicia! response or indicial admittance of the 
system. It gives the displacement of the system, if it is initially resting at 
its equilibrium position and is suddenly subjected to a constant force of 
unit magnitude applied at t = 0. 

From the principle of superposition it can then be seen that the response 
of a system that is initially at rest and then subject to a constant force A 

29 This theorem was also implied in deriving (53). 

30 We assume that the differential equation is of the second order. Otherwise, the 
initial condition shall contain the values of derivatives up to (/? — l)th order. This 
assumption does not affect the generality of subsequent results. 


suddenly applied at t = and to another force B suddenly applied at 
/ = ris 

x(t) = Au(t) + Bu{t - t) t > r > (63) 31 

We assume here that the system is time invariant; that is, the physical 
property of the system does not vary with / or the operator L does not 
contain / explicitly. Otherwise u is a function of two variables u(t, r) 
where r is the time when the force is applied and t is the time when the 
response is measured. u(t) then stands for u(t, 0). Most physical systems 
whose properties vary gradually with time because of aging of components 


Figure 18 

or changes in the environment have become of interest. Such gradual 
changes, however, do not affect their short-time behavior. 

A general time-dependent force F(t), which is applied to a system at 
time t = 0, can be considered as the sum of finite and "infinitesimal" 
step functions, as shown in Fig. 18. Assuming, for the time being, that 

31 The method of defining u(t) previously given leaves it undefined for t < 0, so that 
(63) is valid only for t > r > 0. However, if we extend out domain of definition by 

«(/) = for t < 

then (63) is valid for all values of t and t > 0. For what follows this extension is not 


F(t) is continuous and differentiable for t > 0, we may write 

F(t) = F S(t) + dFS(t - r) 


F S(t) + F'(t) -T(r - t) dr 

By the principle of superposition, if this force is applied to a system that is 
initially at rest at its equilibrium position, the response will be 

f V'(r)«(r - 


x(t) = F u(t) + F'(r)u(t - t) </t (65) 


in which u(t — t) is the response of the system at t due to a unit step force 
applied at r. If «(?) is continuous at f = and differentiable for t > 0, as 
in most problems, we can integrate (65) by parts to yield 

M J 

x{t) = F u(t) + [F(r)u(t - t)] I" 1 - F(t) - u(t - r) dr 

l T==0 Jo AT 

Now w(0) = and (djdr)u{t — r) = — w'(' — t); hence 

x(0 = fV(r) W '(* - t) £/t (66) 


Since the function F appears under an integral sign, the requirements of 
continuity and differentiability of F are not necessary insofar as (66) is 
concerned. 32 Although these requirements were made in deriving (66), 
the same results can be directly obtained by another line of reasoning. 

Consider a time-invariant linear system that is in equilibrium and at rest 
prior to t = 0. At / = the system receives an impulse of force of unit 
magnitude. 33 The effect of this unit impulse is to produce an initial 
velocity x = \\m. With this initial velocity the system is now set into 

32 If the integral in (65) is considered a Stieljes integral, the requirement on F is not 
very stringent. At any rate, in an actual physical problem Fhas at most a finite number 
of discontinuities and points where F' does not exist; the mathematical difficulties can 
easily be overcome by adding the term AFw(r — r^, corresponding to a jump discon- 
tinuity AFat t = t 1} to (65) and splitting the integral to cover ranges where F'(t) exists. 
After integration by parts, the result is still (66), since the terms corresponding to the 
discontinuities in F cancel out just as F u(t) term does. 

33 One may speak of a unit impulse of force at t = as a force described by the Dirac 
function b(t). Although pure mathematicians are still somewhat uneasy about a precise 
definition of such a function, applied mathematicians have accepted it with little qualm. 




motion. Let the response of the system be such that its displacement is 
given by the function h(t). That is to say 

x = h(t) 

l(x) = 


x„ = 




Figure 19 

In a more general discussion we must assume that L is of rath order. 
A unit impulse will then produce the initial condition 

X(\ — Xt\ — 

-(n-2) _ 


~(n-l) _ 

in which a is the first coefficient in L. For the convenience of visualization 
we assume that L is second-order without losing important generalization. 
Figure 19 shows that the action of a force F(t) on a system may be 
considered as that of a series of elemental impulses F(r) dr applied succes- 
sively to the system in the interval < r < t. According to the principle of 
superposition, the response of the system at t is the sum of the responses to 
these impulses. Hence 



F(r)h(t - t) dr 



In comparing (67) with (66), we see that 34 

h{t) = u'{t) 

Thus the displacement response of a linear system to a unit impulse is 
equal to its velocity response to a unit step force. Either way we define it, 
the function u'(t) is called the impulse response of the system. 

In all these derivations it was assumed that at t = the initial condition 
of the system is x = and x = 0. If this is not the case, it is not difficult 
to see that the solution of the problem merely requires additional terms 
corresponding to the nonvanishing values of x and x . 

Let us now find the indicial response and the impulse response of a 
linear damped system. For u(t) we solve the equation 

mx + ex + kx = 1 t > (68) 

together with the initial condition x = 0, x = 0. The general solution 
of (68) is evidently obtained by adding the particular integral \\k to (21). 

x = u(t) = Re (Xe iat ) + - (69) 


It appears that in what follows we assume cjc c < 1 . However, according 
to Exercise 1.11, the results are valid for all damping ratios. 


Since from (27) 


x = Re (A) + - = 


x = Re (iol) = = -Im (aX) 
Re (a) Im (A) = -Re (A) Im (a) 


X = --(1 - /tan (5) = - 


k kcosd (71) 

u\t) = Re (ioXe iat ) = -Im (oXe iat ) 

34 The equality of these two functions cannot really be established merely by com- 
paring (66) and (67) because / appears both in the integrands and in the limits. Never- 
theless, these functions can be shown to be identical. To prove this fact, utilize the 
formula to be proved in Exercise 1.32 and remember that F(r) can be any arbitrary 


Utilizing (22), (27), and (71), we have 

CO 2 

u(t) =— - e-< cl2m)t sin co c t 



Derivations for the same relationships by the classical method are as 
follows (see also Exercise 1.24): 

x = u{t) = Ce~ {cl2m)t cos (aj c t - a) + - 


M (0) = C cos a = - 

u\t)= -Ce- {cl2m)t 


cos (co c t — a) + co c sin (co c J — a) 

u'(0) = — cos a = ca c sin a 

division of the equation by w and the application of the relations in (27) 

1 CO 

a = -5 C = - = - — 

k cos a kw n 


koj n 


u(i) = U 1 - — e - (cl2m)t cos (oj c t - S) 
k\ co c 


Equation (72) can be obtained much more readily if we solve for the 
problem of free vibration due to an initial velocity x = 1/m. 35 
In that case we take the general solution (21): 

Also as 


x = h(t) = Re (Xe iat ) 
x = so Re (X) = 

^ = Re (iaX) = -Im (gX) = -Im (A) Re (a) = — 


Re (o - ) = co c Im (X) = 

35 On the other hand, the reasoning we used in deriving (67) is not mathematically very 
rigorous, although such rigor can be had by refining some of the arguments. 




moj c 


A(0 = Re — e iat ) = m'(0 (73) 

\ moj r J 

It can easily be verified that (73) is the same as (72). The foregoing results 
contain as a special case the indicial response and impulse response of an 
undamped system. For such a system let 

c = b = 

sin cot 

CO = 

oj c = 

= a 

c = 

Thus (69) and (72) become 

u(t) = 


■ cos cot) 

u'(t) = 


- — sin o)t = 


sin cot 


Substituting (72) into (66), we have the solution for forced vibration of a 
damped system, which is initially resting at its equilibrium position. 

If the system has an initial displacement and velocity, complementary 
solutions (19) with the constants determined in (23) must be added. The 
complete solution is therefore 

x(t) = e-«l**»(x cos co c t + Cc± « + CMX ° sin co c t) 

\ OJ r C r J 

co c c c 

P CO 2 

+ —7 e- {cl2mKt - T) sin co c (t - t)F(t) dr (75) 
Jo co c k 

For an undamped spring-mass system, with c = and co c = to, the 
solution becomes 

x(t) = x cos cot H — - sin cot + — sin co(t — r)F(r) dr (76) 
w Jo k 

In summary, the particular solution satisfying a linear differential 
equation with constant coefficients and a set of initial values 

\_(x) = F(t) when / = x = x x = i 

etc., is made up of two parts; one part represents the motion 36 which 

36 We use the words "motion" and "force" merely to fit the physical phenomenon 
under discussion. The theorem is, however, strictly mathematical and can have other 




would prevail if F = 0, and the other part represents the motion caused 
by the force F{t) acting on the system subsequent to t = 0. 37 In contrast, 
the classical method illustrated in Arts. 1.7 and 1.9 obtains the general 
solution to the differential equation first; this consists of a complementary 
function, representing the transients, and a particular integral, represent- 
ing the steady-state motion. 38 The particular solution, when needed, is 
then obtained by determining the constants in the complementary function 
to satisfy the initial conditions. When the applied force is periodic, and 
only the long-term behavior of the system is of interest, this method of 
solution is a natural one. Conversely, when the applied force is of short 
duration and only the short-term behavior of the system is wanted, the 
method described in this article is more natural. Nevertheless, as far as 
mathematics is concerned, either method can be used in either case. 

1.11 Signal-Response Relation 

of Linear Systems in General 

The problem we have studied in the last few articles was formulated in a 
restricted way. We sought the force-displacement relationship in a spring- 
mass-damper system of a certain arrangement. The essential part of the 
analyses has, however, much more general applications. 







Figure 20 

Let us visualize a spring-mass-damper system enclosed in a "black box," 
as shown in Fig. 20. The three weightless rods, A, B, and C, which are 
attached to the components of the system, protrude through the walls of 
the box and can either slide without friction or be locked to the wall. The 
problem investigated was, "What kind of displacement will be produced 

37 In this approach the condition of the system prior to / = is not considered at all. 

38 The terms "transient" and "steady-state motion" are devoid of physical significance 
when the applied force is not aperiodic function of time, although the two parts of the 
solution are commonly so designated. 




in A by a force applied at A, provided that B and C are locked to the box, 
which is held fixed?" In other words, "How does the black box transform 
a force applied at A into a displacement of A ?" Evidently, this is not the 
only behavior of the box that could be of interest. For instance, one may 
investigate, "What is the force transmitted to the foundation, to which the 
box is attached?" Or one may change the operating condition of the 
system by loosening the lock on B and moving B by some external means, 
and then ask, "What will be the motion transmitted to A ?" We can also 
ask the same questions about a more complex system, such as that shown 
in Fig. 21. In all of these cases an external disturbance or excitation, 
such as a force, an impulse, or a displacement, is imparted to the system 

mms^z^^&m^^mimsmmzEEzzzzzzzzzzzz^^^^ EZEz^z^Ezzmzzzzz 

Figure 21 

at a certain place, and we look for a certain resulting change in the system 
that interests us. This excitation, transient or periodic, is called a signal, 
and the change in the system that interests us is called the response to the 
signal. Clearly, the designation of a particular change as the response is 
purely a subjective matter. What we wish to emphasize here is that many of 
the analytical results obtained in the preceding articles can be immediately 
generalized to cover the all-signal-response relationships in linear systems. 


Consider the superposition theorem described by (66) and (67). Since 
in the derivation of these equations no reference is made to the exact make- 
up of the systems or to the nature of the signal and the response, it is 
evident that they apply to all linear systems with time-invariant components. 
Take, for instance, the question of the force transmitted to the foundation 
of the box of the system in Fig. 20. If F(t) is the force applied at A and 


/(/) is the force transmitted to the foundation, the problem can be formu- 
lated by the following set of differential equations: 

mx + ex + kx = F(t) , 

ex + kx=f(t) 

Although it is possible to eliminate the variable x in the foregoing set of 
equations to obtain a single equation in the standard form of 

mf + cf + kf= cF+ kF= G{t) (78) 

it is not necessary to do so. 39 According to our previous reasoning, if 
u f (t) is the force transmitted to the foundation when a unit step force is 
applied at A, the corresponding force transmitted with F(t) is given by 

fit) = P »/ 


(t - r)F(r) dr (79) 

provided that the system is initially at rest and in equilibrium. 

To evaluate u f (t), we can first evaluate u x {t), which is the solution to 
the equation 

mx + ex + kx = S(t) with x = and x — 
and then utilize the relationship 

cu x {t) + ku x (t) = u f (t) 

If we are to analyze the same problem for the system in Fig. 21, the 
indicial response function u f (t) will be a more complicated one and will 
have to be obtained by methods discussed in later chapters; but when it is 
found the rest of the procedures will be the same. 

The most important lesson to be learned from this discussion is that the 
dynamic property of any linear system is completely specified by its 
indicial or its impulse response. These functions describe how a system 
behaves under any given excitation or how it will change any given input 
signal into an output. In short, they describe the system just as completely 
as the differential equation itself. Furthermore, for many actual systems, 
because of the uncertainties in the measurement of such parameters as 
masses and spring constants, the differential equations themselves are 
likely to be less reliable than the indicial response functions when the 
latter can be measured directly. 


Another way of describing the dynamic property of a linear system is 
by its steady-state response to sinusoidal inputs. This description is 

39 See also Art. 1.17. 


furnished by the so-called frequency response of the system, that is, the 
amplitude and phase difference (lag) of the response as functions of the 
frequency. Take, for instance, the system shown in Fig. 20. Let the 
signal be in the form of a sinusoidal motion imposed on B, and let the 
motion of A be considered as the response, with C clamped to the box. 
The differential equation relating the displacements of A and B is 

mx a + cx a + k(x a — x b ) = 
Upon letting 

x h = \).,\ COS CO f t 


x a = \?. a \ cos (m f t - x) 

a comparison with (38) shows that the amplitude ratio |/- a |/|/. 6 | and phase 
lag a are the same as the magnification factor 40 \/.\ld st and angle a defined 
in (41). 

It is more convenient, however, to use complex-number representation 
to describe the signal to response relation so that both the amplitude ratio, 
generally called the gain, and the phase lag are combined in one expression. 
According to (46), the frequency response in this case is given by the com- 
plex number 

§ 'a 



~(-) 2 + 

£0 ' 

in which / a and ). h are then the complex amplitudes of A and B. respectively. 
Consider now the relation between the force transmitted to the founda- 
tion /and the force applied to A, that is. F. According to (78), if we let 

f=& m > % and F=F e^ 
we have 

/ icco, + k 

F k — mcof + ieco f 

= T r (ico f ) (80) 

The absolute value represented by this ratio, or the gain, is called trans- 

f " N [C <" (81) 

"i _ te)T+ p5r)' 

V CO 1 J V c c CO ' 

40 In the present connection the name of this factor gives a more graphical description 
to its physical nature. 




Figure 22 

This ratio is plotted in Fig. 22 and is of practical interest in the design 
of supporting structures for machines susceptible to vibration. 


In general, if X r is the complex amplitude of the response and X s is that 
of the signal, the ratio between the two is a complex function of variable 
io) f , which describes the frequency response of the system 

l r = T{ko f )X s (82) 

The two examples show that the function T is a ratio of two poly- 
nomials of ico f with real coefficients, which are the coefficients in the 


differential equation. Although in our discussion ioj f is always imaginary, 
to study the analytical properties of this function, hence of the differential 
equation itself, it is sometimes advantageous to extend its domain of 
definition to allow complex argument. Because T is a rational function 
this extension, by replacing the pure imaginary argument, uo t with a com- 
plex argument s, produces function values of T which "blend smoothly" 41 

f' -mo) f 2 + icw f + k 

Im [T(ico f )] 

Figure 23a 

into the values in its restricted domain of definition. The utility of the 
function T(s), called the transfer function, is connected with the theory 
of Laplace transformation, a powerful tool in analyzing linear systems, 
which, unfortunately, cannot be discussed at length here. It is mentioned 
only to introduce the reader into a casual but logical acquaintance with a 

41 Without entering into the theory of analytical continuation of analytic functions, 
this is the best we can do to describe this process. The statement is intended merely to 
convey some rough idea of a process which is not of primary concern to us here. 





subject that, it is hoped, he will study more intensively elsewhere 
remark applies also to the rest of the discussion under this heading. 

To represent the function T(ko f ) graphically, we have been plotting its 
absolute value (or the gain) and its argument (or the phase lag) as functions 
of o) f . This method results in two curves. There is another method of 

T(s) = 

ms z + cs + k 

0.2 0.4 0.6 0.. 
Figure 23 b 

1.0 1.2 1. 

1.6 1. 

graphical representation which cross-plots the two to give a locus of T in 
Argand's diagram with co f as a parameter. Such a plot is called the 
Nyquist diagram or Nyquist locus. The Nyquist locus, representing the 
frequency response of a simple mass-spring-damper system, is shown in 
Fig. 23a. Figure 23& assembles a number of such loci with different values 
of cjc c . Consider T(s) as a function of a complex variable s, the Nyquist 
locus is then the "map" of the imaginary axis of the s-plane on the T- 


The generic term transfer locus is used to include the Nyquist locus, T, 
the inverse Nyquist locus IjT, and the over-all transfer locus Tj(\ + T). 
All are useful tools in the analysis and design of linear control systems. 

(d) relation between the indicial response 

and the frequency response of a linear system 

We have seen that both the indicial response function and the transfer 
function can be derived from the differential equation describing the 
system. We have also shown that the indicial response by itself describes 
adequately the dynamic properties of a system. We now proceed to show 
how the indicial response function is directly related to the transfer function. 
This relationship, when established, serves two purposes. Analytically, it 
proves that the frequency response also gives an adequate description of 
the system. Experimentally, it furnishes a better way of determining the 
dynamic characteristics of a system because the measurements of the 
frequency response, which are those on a steady-state phenomenon, are 
easier to make than those on a transient one. 

It is necessary here to digress into a mathematical formula to be used 
later. Any standard textbook 42 on advanced calculus contains the evalua- 
tion of an improper integral 

!*% = * 

6 2 



Upon replacing 6 by co f t, it is not difficult to see that 

f x sin co f t , 

f -dco f =\ 

Jo c>, 

/ = 

The step function _T(f) can then be expressed in terms of this improper 
integral as 

1 1 f °° sin (oJ , , M ^m* 
-T(0 = - + - f -dco f (83) 13 

2 77 JO CO f 

42 See, for instance, Sokolnikoff, Advanced Calculus, McGraw-Hill, New York, 1939, 
pp. 361, 362. 

43 Strictly speaking, this equation is not valid at one point, viz., t = 0, when the 
left-hand side is defined as 1 and the right-hand side is \. This difference is, however, 
of no consequence here. 


The indicial response u(t) to a linear system represented by the differen- 
tial operator L can thus be considered as the solution to the equation 

(/ x 1 1 f 00 sin ay J 

L 77 J0 COf 


It must be pointed out that replacing (62) by (84) really involves a 
change in physical considerations as well as in the mathematical descrip- 
tion of the problem. The step function S(t) was previously chosen to 
characterize a nonperiodic excitation. Although the function value was 
also defined for t < 0, its definition for negative values of t is not essential 
because our analysis covers only the time period beginning at t = when 
w (0) = w'(0) = . . . = ^"-^(O) = 0. On the other hand, physical reason- 
ing clearly shows that if we extend the time axis of analysis backward to 
t = -co the fact that S(t) = for all negative /'s means that the system 
has been completely free from excitation since time immemorial and 
consequently it must be in a completely quiescent state at t = 0. 44 There- 
fore, by utilizing the definition of a unit step function for negative values 
of /, we have altered our consideration of the indicial response from that 
due to a nonperiodic excitation to that due to a periodic excitation of 
infinite period, which, according to (83), is the sum of sinusoidal functions 
of all frequencies. Each frequency contributes an amount equal to dco f /7TCo f , 
and the contribution of zero frequency is the constant h 

Let the transfer function associated with the differential operator L be 
T(s) and the real and imaginary parts of the frequency response function 
T(ko f ) be R and /, respectively, 

T(ico,) = R(co f ) + U(o),) 

Equation (82) can then be written as 

l/^ = [R( j f ) + /Y( W/ )] A/^ 

Let k s = 1, X s e lMft = cos co,t + i sin co f t. 

Re (X r e iu} ^) = R(oj f ) cos co,t — I(co f ) sin co,t 

Im (h r e 1UJft ) = R(co f ) sin co,t + 1(a),) cos o f t 

According to the discussion in Art. 1.7, the right-hand sides of the 
last two equations are the responses to signals equal to cos co,t and sin 

44 In essence, we are assuming that the system is stable and has energy dissipation 


(o f t, respectively. Hence we have the following signal-to-response relation- 

Signal or Excitation Response 

i = i cos 0/ i [R(0) x cos Ot] = iR(0) 

sin co f t R(oj f ) sin oj f t + I(oj f ) cos oj f t 

-T(t) u(t) 

The solution to (84) can thus be obtained through superposition: 

u(t) = \R(0) + - [/tfw,) sin co f t + /(a),) cos co f t] — (85) 

7T Jo 0)^ 

If the resulting improper integral converges uniformly with respect to ?, we 
may differentiate (85) with respect to t to obtain 

1 f 00 
u'(t) = — [R(oi f ) cos oy — I(co f ) sin w^] ^/c/^ (86) 

77" JO 

Before we proceed, it must be realized that the operations leading to 
(85) and (86) are strictly formal; that is to say, there is a number of 
mathematical questions that must be settled before we can accept the 
results as legitimate. The subject belongs to the theory of Fourier trans- 
forms, which is not within the scope of this book. We can, however, deal 
with the more essential points here. 

Let us introduce the symbol L _1 to denote the operation that yields the 
particular integral of a linear differential equation: 

L(*) = F(t) 
x = L-iF(0 

This inverse operation can be carried out by a number of methods, 

including such procedures as Lagrange's variation of parameters. We do 

not have to be concerned with the exact method used, except that essentially 

it involves integration processes. The solution to (84) should be of the 


r, , J l \ l , i f 00 sin ay , 
u(t) = L- 1 - 1 + - L-i L da> f 

\2! rr Jo tO f 

whereas (85) is in reality 

/1\ 1 f 00 , JsmG) f t\ , 

Since L -1 involves integration with respect to /, the validity of (85) depends 
upon whether the order of the two integration processes can be reversed. 


For the integral in question the sufficient condition for the legitimacy of 
order reversal is that the integral should be uniformly convergent for any 
finite ranges of t, — oo<a<t<b<co. Upon examining the inte- 
grand, we see that uniform convergence is assured if I(co f )jcOf has a finite 
limit at co f = 0, I(co f ) approaches at co f = oo, and the absolute value of 
T(ico f ) is always finite. The first two conditions are satisfied when I(co f ) is 
the imaginary part of a transfer function. (See Exercise 1.34.) The last 
condition is satisfied when the system is damped and has no resonance 

However, with more sophisticated mathematical tools, (85) and (86) 
can be derived without the requirements that T(ico f ) be bounded and that 
the integrals converge uniformly. Instead, these integrals are interpreted 
as the limits of a sequence of convergent integrals containing a parameter 
a as a approaches 0. Physically, the procedure is equivalent to introducing 
an extra damping force so that the resonance condition cannot take place 
and then letting this force approach zero in the solutions obtained. If the 
limiting integral converges to a so-called Cauchy's principal value, the 
result is then valid. 

Returning now to (85) and (86), we observe that u(t) and u'(t) should be 
identically zero for / < 0. Hence, if Ms a positive number 

= «(— t) = - R(0) + - [— R(co f ) sin co f t + I(co f ) cos co f t] — - 

2 77 JO 0) f 

Combining this equation with (85), by addition and by subtraction we 

u(t) = R(0) + - I(oj f ) cos co f t — (t > 0) 

77 JO (Of 


u(t) = - ( R(w f ) sin co J -^ ) (t > 0) (87) 

77 JO \ (O f J 

Similarly, u'{—t) = 0, if t is a positive number. 

1 f 00 
= - [R(co f ) cos co J + I(co f ) sin co f t] dco f (t > 0) 

77 JO 

Together with (86), this relation gives 

2 f 00 
u'(t) = — R(co f ) cos co f t dcOf 

77 JO 


2 poo 

= I(cQf) sin co ft dcOf (t > 0) 

77 JO 


The expressions (87) and (88) are valid only for t > 0. When the 
transfer functions are determined from the frequency response measured 
experimentally, the integrals in these expressions can be evaluated by a 
numerical procedure or by mechanical integrating devices. 45 When the 
transfer function is known analytically, these integrals cannot be con- 
veniently evaluated as they stand. It is usually necessary to carry out 
the integration process in the complex plane by the following trans- 

First, let us write (86), which is valid for all t, as 

If 00 
u'(t) = - Re [T(ico f y a ' t \ d<o f (89) 

7T JO 

Now consider the function 

T(ico f )e^ = T(ico f )(l + ia> t t + ^ + ( -^ r* • • •) 


in which T(ico f ) contains only real constants, except for the argument ko f 
itself. Since i and co f enter together as a product in this function, its real 
part is an even function of a> f and its imaginary part is an odd function 
of co f . Hence 

f °°Re [T{io) f )e im f l ] dco f = I Re [T(io) f )e i<0 ^\ da> f (91) 

Jo J -co 


f °°Im [Tiico^e^] dco f = - f Im [T(ia> f )e ito ' t \ dco f (92) 

Jo J - oo 

We can then write (89) as 

1 f 00 
u\t) = - Re [T(iQ) f )e %m ^] dco f 

77 Jo 


Re [T(iw f )e iai S] dco f (93) 


[T(ico f )e %m ' t ] dco f 

The sign "Re" was dropped in the last integral, since its imaginary part 
will vanish because of (92). Replacing the dummy variable ico f in (93) by s 
puts the integration in the complex plane, and we have 

w'(0=^-. P°° T(s)e st ds (94) 

Z777 J-ioo 

45 "Determination of Transient Response from Frequency Response," by A. Leonhard. 
ASME Trans. Vol. 76, 1954, p. 1215-1236. See also discussion of the paper by A. 


This is an important relationship in operational calculus. It states that 
the impulse response is the inverse Laplace transform of the transfer 
function, or that the Laplace transform of the impulse response is the 
transfer function. Ordinarily, the relationship is written 

i rc+io 

.77/ Jc—icc 

u'(t) = — T(s)e st cis (95) 

2.7TI Jc—icc 

The constant c allows a shift in the path of integration from the imaginary 
axis to another line parallel to the imaginary axis without changing the 
value of the integral. 46 Such shifts may facilitate the evaluation of the 
integral and avoid the mathematical difficulty when the system has 
resonance conditions and T(ico f ) becomes infinite for certain values of 
ico f . Similarly, (85) may be transformed to an integral in the complex 
plane as 

1 (*c+i co e st 

u(t) = — T(s) - ds (96) 

■^TTl J c — ico S 

This integral is associated with the name of Bromwich. 

As a practical matter, let it be pointed out that there are extensive 
tables available which give the pairing of functions of t with functions 
of s through Laplace transformation or its inverse. These tables, together 
with the linear characteristics of the operation, make it unnecessary to 
carry out actual integration processes for the evaluation of these integrals. 

The primary purpose of this discussion is to show that there is a direct 
connection between the steady-state response and the transient response. 
In the process of showing this connection, we have taken a peek into the 
subject of operational calculus, which is the most powerful tool for 
transient analyses of linear systems. This subject is, however, too large 
and important to be included in this book. Moreover, from the viewpoint 
of applications, elaborate analysis of transient phenomena is seldom made 
in connection with mechanical vibrations, as it is with control systems and 
instrumentations, which form courses of study by themselves. 

In concluding this article let us make one more generalization. At the 
beginning we chose as a model a signal-to-response relationship governed 
by an equation of the type 

L(z r ) = x s (97) 

in which L is a differential operator with constant coefficients. There is 

46 The theory of functions of a complex variable has precise things to say about how 
much of a shift is permissible. 


nothing really to prevent us from dealing in exactly the same way with the 

l ± (x r ) = L 2 (x s ) (98) 

or with a set of n linear equations in the general form of 

IW= L i( x s) i= 1,2, •••,/! 


with one of the x's at the left-hand side chosen as the response. In other 
words, we have never restricted our reasoning to the case of single-degree- 
freedom systems. Consequently, with minor modifications, the results 
obtained can be applied immediately to systems with multiple degrees of 


1.12 Examples of Linear Vibratory Systems 
with a Single Degree of Freedom 

It has been pointed out that the systems shown pictorially in Figs. 1 
and 5 are merely models of a class of physical systems whose outward 
appearances may be quite different. In this article we give a few such 
examples. 47 For this purpose it is necessary that we set up only the 
differential equations governing the motions of these systems and show 
that they are of the form of (2) or (14). The solutions to these equations 
have already been studied. 

(a) torsional vibration of a disk on an elastic shaft 

Let an elastic shaft be fixed at one end and carry a disk of moment of 
inertia /at the other end, as shown in Fig. 24. Assume that the mass of the 
shaft is negligible. The configuration of the system can then be described 
by the angle of twist 6. Because the shaft is elastic the torque exerted by 
the shaft on the disk must be proportional but opposite in direction to the 
angle 6. Hence 

Id = -k6 or Id + kd = 

where k is the torsional stiffness of the shaft. 

(b) oscillation of a liquid 

in a u-tube of uniform inside diameter 
Let the U-tube shown in Fig. 25 be in a vertical plane and filled partially 
with an incompressible liquid. The liquid levels at the two sides of the 

47 For other examples see Arts. 1.15 and 1.16. 


tube may be set into an oscillation by momentarily applying a pressure to 
one side of the tube and then releasing the pressure. Let 2x be the differ- 
ence in the levels of the two sides. The potential energy of the system is 
equal to that of elevating a liquid column of length z to a height x. Hence 

V = pgAx 2 

in which p is the density of the liquid and A is the cross-sectional area of 
the tube. The kinetic energy of the system can be seen to be 

T = ipALx 2 







Figure 24 

Figure 25 

in which L is the total length of the filled tube. If the viscous and capillary 
forces are neglected, we have 


(T + V) = 


Lx + 2gx = 

If the viscosity has to be taken into account and a laminar flow can be 
assumed, the viscous force per unit length can be obtained from the well- 
known Hagen-Poiseuille law as 

dF Sjux 

~dl = ~i r 

in which /u is the dynamic viscosity of the liquid and r is the inside radius 
of the tube. The differential equation is then 


pALx + 

+ IpgAx = 





X + 

lv 2? 

— x + -2-x = 

trr* L 

in which v is the kinematic viscosity. 


The system shown in Fig. 26 consists of two springs and a damper 
attached to a step pulley that rolls withoutslipping on a rough surface. 



Figure 26 

It is a system of no practical utility and is conjured up merely to illustrate 
our method of analysis. 

The motion of the pulley can be described by the displacement of the 
center mass, x, and the angular displacement, 0. Because it rolls without 

Rdd = dx Rd = x + constant 

in which the integration constant is zero if both 6 and x are taken to be 
zero at the equilibrium configuration of the system. The system has 
therefore only a single degree of freedom, and either 6 or x can be chosen 
as the coordinate variable of the system. 

Let M be the mass of the pulley, /, its moment of inertia about the 
center of mass, and F, the frictional force at the contact. The two 
momentum equations, linear and angular, are 

Mx = -k x x- k 2 (x + rd) - C(x - rd) - F 

16 = FR + C(x - rO)r - k 2 (x + rO)r 

Upon substituting x by Rd and eliminating Ffrom the foregoing equations, 
we have 

(MR 2 + 1)6 + C(R - rfd + [k ± R 2 + k 2 (R + rf]d = 


or, if we prefer, 

(MR 2 + I)x + C(R - rfx + [^R 2 + k 2 {R + r) 2 ]x = 

They are in the standard form of 

mx + ex + kx = 

Let us now analyze the problem by energy considerations, which leads 
to the observation that 

Ut + V) + P = 

in which P = power dissipated in damping. For this system 

T = \Mx 2 -+ \I0 2 = \{MR 2 + I)6 2 

V = ik ± x 2 + ik 2 (x + rQf 

= \[k x R 2 + A: 2 (/J + rf]d 2 

P = C(R- rfd 2 

(M/? 2 + 1)68 + C(,R - rfd 2 + [^/? 2 + A: 2 (/2 + r) 2 ]06 = 

Since d ^k 0, it can be canceled out and the same equation is obtained. 

1.13 Linearization of Systems in Small Oscillations 

Very few physical systems are strictly linear. On the other hand, a 
large number of them can be so considered if they are in motions represent- 
ing only small changes from their equilibrium configurations. A general 
discussion on this subject is taken up in Art. 3.2. At present we shall 
study a few examples. 

(a) simple and compound gravity pendulums 

It is well known that the motion of a simple pendulum (Fig. 21a) is 
described by the equation 

LB + g sin 6 = 

in which L is the length of the pendulum, is the angular displacement of 
the pendulum from the vertical, and g is the gravitational acceleration. 
Obviously, this differential equation is not linear. Its solution is in the 
form of an elliptical integral. On the other hand, if the swing of the 
pendulum is limited to small angles, then sin 6=6, and the equation may 
be approximated by 

+ |fl = 


Hence the period T of the pendulum is approximately 

T = — 




The period of a compound pendulum (Fig. 21b), having a mass M, a. 
radius of gyration about its center of mass r, and a point of suspension at a 
distance d from its center of mass, can be found most conveniently by 
Rayleigh's method if its angular displacement is small. Let 8 m be the 
amplitude of the swing, which is the maximum angular displacement, and 
6 m be the maximum angular velocity, which takes place at = 0. The 




Figure 27 


maximum kinetic energy and the maximum potential energy of the system 
are then 

r mas = \M{r 2 + d*)dj 

^max = Mgd{\ - COS 6 m ) 

For small m , 1 — cos B m = \0 m 2 ; the system is approximately linear and 
its motion, approximately simple harmonic. Hence 

6 2 = a> 2 2 

Upon equating the two energy expressions, we have 



r 2 + d 2 


r 2 + d 2 

(b) a spring-loaded pendulum 

In Fig. 28 a tension spring is attached to an otherwise simple pendulum 
of length la. One end of the spring is fastened to a point at a distance a 
above the suspension point of the pendulum. The other end is attached to 


the mid-point of the pendulum. The spring constant is k and its natural 
length is a. Let us assume that the pendulum is heavy and the spring is 
soft so that the equilibrium position of the pendulum is the vertical. In 
terms of angular displacement 6, the energy expressions are 

T = im(2adf = \(<\ma 2 d 2 ) 

V = 2mga(\ - cos 0) + |£(AL) 2 + C 

Figure 28 

in which AL is the change in the length of the spring from its natural 
length and C is a constant that will make V(0) = 0. 48 

AL = V2a 2 + 2a 2 cos 6 - a = ail cos - - l) 

At = 0, AL = a, hence C = —ka 2 \2. By substitution and simpli- 
fication, we have, finally, 

V = 2mga{\ — cos 0) — 2ka 2 cos -I 1 — cos- 1 

Now, for small oscillations 

1 — COS = — cos - 

2 2 

e e 2 

1 — cos - = — 

2 8 

V = \{2mga - ika 2 )0 2 

48 Potential energy expressions are determined only to within an additive constant. 
In order to use Rayleigh's method, this constant must be such that V vanishes at 
equilibrium configuration. 


Rayleigh's method then yields 

2mga — ka 2 jl j g k 

4ma 2 v 2a 8m 

It is interesting to note that the spring in this system exerts an influence, 
as if it had a negative spring constant, because its potential energy is a 
maximum instead of a minimum at 6 = 0. In order for the system to be 
stable at = 0, it is necessary that 

ka 2 " 
2mga > — 

Otherwise co becomes imaginary, and no oscillation around 6 = is 
possible. As a matter of fact, 6 = then would not be a stable equilibrium 
configuration. On the other hand, as long as this inequality holds, it is 
possible to obtain a very "soft" system with a very low natural frequency. 
This scheme of having two restoring forces working against each other to 
obtain the effect of a soft spring has practical applications, since a single 
soft spring is usually inconvenient to incorporate in a mechanism. 


The foregoing examples belong to a class of nonlinear systems whose 
differential equation of motion is of the type 

mx + K(x) = 

where K(x) is an odd function that may be represented by 

K(x) = kx + k'x 3 + R 

The remainder R is of the order x 5 . The corresponding expression for 
potential energy is then 

V = K(x) dx = - kx 2 + - x* + \R dx 

As a first approximation, let us take 

K(x) = kx and V(x) = \kx 2 

The frequency w is then Vk/m. To get a better approximation, we can 

K(x) = kx + k'x 3 


V(x) = \kx* + \k'x' 


The first integral of this second-order differential equation is 

V m = T+V T=V m -V 



mx* = k(xj - &) + -k'(xj - a*) = k(xj - x>) 
From this we have 
dt = 


*J + * 2 )] 


Vk(xJ - x 2 )/m I ,{ 


Vk(x m * - v*)\m 

[ X +2k {X - +x2) 
[l -£*? + * 

If this equation is integrated between the limits x = and x = x m , the 
left-hand side becomes the quarter period 

T = jT___k^_ p» (xj + 
4 2co 4co& Jo Vx 2 - 

a; 2 ) 

— 1 x, 

2co\ %k ' 

Vx^ — X' 



in which co = Vk/m is the frequency in the first approximation. The 

2n t 



can be considered as a second approximation. To check its accuracy, we 
take the case of a simple pendulum swinging a total angle of 60°. 

T= V(Z/i)2^1 + ^) = 6.39lVZfe 

r 2 

The exact answer is 

= W^J"' 


VI - sin 2 15° sin 2 
For a 120° swing the formula is in error by 3 per cent. 

= 6.392VL/g 

1.14 Piece wise-Linear Systems 

The type of system to be discussed in this article has vibratory motions 
that are governed by several linear differential equations, each of which is 


applicable only for a certain range of values of displacement or of velocity. 
Strictly speaking, such a system is not linear, since it has few of the 
important properties of linear systems. Its analysis is usually difficult. A 
few simple cases are introduced here to give us some feeling of such 

(a) a system with an unsymmetrical restoring force 

In Fig. 29 a mass m vibrates between two compression springs k ± and fc 2 . 
The natural lengths of the springs are such that only one of the springs 


i\/vv\/vwwvf^~ " Vwvvwwvww 

Figure 29 

is under stress at any given time. The differential equations governing the 
motion of m are then 

mx + kjX = for x > 

mx + k 2 x = for x < 

To solve these equations, let us assume x = and x = i > at 
t = 0. For the time being, we shall look only for a solution that is valid 
for a certain time interval containing the instant t = 0. It is not difficult to 
see that 

x(t) = Q sin cojt for < t < t ± 

x(t) = C 2 sin co 2 t for > t > t 2 

could be made to satisfy the equations of motion and the initial condition, 
at least for the certain time interval t 2 < t < t v Since x Q is positive, 
co ± = Vkjm and a> 2 = Vk.Jm. Since velocity must be a continuous 
function of t, at t = 0, 

Xq — COiCi — CO oL 2 


r> _ ^° r — ' T ° 

c x — — c 2 — 

The limits f x and r 2 are determined by the condition that x{t) should not 
change its sign in < t < t l9 nor in > t > t 2 . Hence 

7T 77 

CO, CO « 


The solution is therefore 

x(t) = — sin corf for < t < 


x(t) = — sin aj 2 t for < t < 

Obviously, the motion is periodic with period 

T=t 1 -t 2 = 7r(- +-) 
\co 1 0) 2 / 


Figure 30 

The solution outside the fundamental interval can be expressed as 

x{t) = — sin co^t - nT) 

if there is an integer n satisfying 

< co^t - nT) < 77 

x(t) = — sin co 2 (/ — nT) 
a> 2 

in which n is an integer satisfying 

— 7T < co 2 (t — nT) < 

A graphical representation of the solution is shown in Fig. 30. 

Although the solution for free vibrations of this nonlinear system is 
comparatively simple, that of forced vibration is algebraically complicated. 


Suppose the system is subjected to a harmonic force F cos (oj f t — p) and 
its initial condition is x(0) = 0, x(0) = x > 0. The differential equation 
valid for a certain time interval < t < t t is 

mx + kjX = F cos (co/ — fi) 

The solution of this equation can be written as 

x(t) = Q cos (oj ± t — a x ) + A x cos (a> f t — ft) 



-ma) f 2 + Atj 
Q cos a x + A x cos P = 


i n — /l, o) f sin tf 
tan a x = / / 

— /t^ COS p 

The upper limit t x of the time interval for which the solution is valid is the 
smallest positive t x that satisfies 

x(t^) = C x cos (co^ — a 2 ) + A x cos (co f t — /3) = 

xitj < 

In conjunction with the condition x(0) = we have 
cos a x cos /? 

cos (w^! — a x ) cos (cofa — P) 


Solve for the smallest positive root of t x in this equation and obtain 
x x — x(tj). Use this as our new initial condition and repeat the process for 

mx + k 2 x = F cos [co f (t — r x ) — f3] 

x(t^) = x{t^) = x x t x < t < t 2 

This example shows that for a nonlinear system we cannot always speak 
of transients or steady states, even when this system is capable of free 
periodic motion and the applied force is a periodic force. 

(b) a spring-mass system with coulomb damping 

In Coulomb damping the damping force is constant in magnitude but 
always in a direction that resists motion. When there is no motion, its 
magnitude may assume any lower value that is enough to resist impending 
motion. Such damping force is an idealization of forces due to dry friction. 




Suppose we have a simple spring-mass system subject to a Coulomb 
damping force of magnitude/, as shown in Fig. 31. Let x be the displace- 
ment of mass m from its equilibrium position in the absence of/. The 
vibration of this system is then governed by the following set of equations: 

mx + kx = — / 
mx + kx — f 

for x > 
for x < 

Or more concisely, 

mx +/sgn (x) + kx = 

The symbol "sgn" stands for "sign of." It denotes a function that takes 





/ i 


Figure 31 

the value +1 or — 1, according to whether the argument is positive or 
negative. 49 Hence 

sgn(0) = 0/|0| 

It is also customary to define sgn (0) = 0, although in our case the 
friction force is undefined when x = 0. The last way of writing the 
equation reveals more clearly the nonlinear nature of the system, but it 
is more convenient to deal with our original way of writing two equations 
in solving the problem. 

Without the loss of generality, let us assume that the system is set into 

49 The second term of (83) is such a function. Also, the differential equation of motion 
for the system studied in (a) of this article can be written as 

mx + 

(k x + k t k x - k 2 

s g 

n (x)\x = 


vibration by giving the mass a positive displacement D and releasing it. 
The initial condition is then 

x(0) = D and x(0) = 

Let us assume further that 



The restoring force kD is thus greater than the friction force/, and in the 
initial stage the motion will have a negative velocity. The governing 
equation during a certain time interval < t < t x becomes 

mx + kx =f 

x -f- oj 2 x = co 2 d 

Its solution under the given initial condition can be seen as 

x = {D — d) cos cot + d for < t < t Y 

The vibration is therefore a simple harmonic motion about the point 
x = d, with amplitude D — d, and circular frequency to. This solution 
is valid only as long as 

x = — oj(D — d) sin cot < 

Hence r x = tt\ cq, which is the half period of the free vibration. At t = t lt 
x x = 2d — D. If \x x \ > d or D > 3d, the restoring force kx 1 is again 
greater than the friction force /, and the mass will eventually acquire a 
positive velocity. The subsequent motion is governed by 

mx + kx = —f 

x + co 2 x = —co 2 d 
together with the condition 

xitj) = D - 2d and ac('i) = 

For the time interval t x < t < t 2 the solution becomes 

a; = (D — 3d) cos cot — d 
in which t 2 = 27r/co. 

Repeating this procedure, we perceive that the motion can be described 
by patching together simple harmonic motions all having the same 
frequency co, which is the natural frequency of the spring-mass system. 
Each of these motions lasts only for half a cycle during which the velocity 
does not change sign. After each half cycle the amplitude decreases by an 
amount equal to 2d = 2f/k, and the mid-point of the cycle shifts from 




x — d to x = — d, or vice versa. The motion stops at the end of a half 
cycle whose amplitude is less than 2d. A graphical representation of the 
motion is shown in Fig. 32. 

The forced vibration of a system with Coulomb damping is difficult to 
analyze. An exact analysis by J. P. Den Hartog is too elaborate to be 
given here. We present instead the following approximate analysis. First, 
let us observe that if the Coulomb damping force is only of moderate 

stops at 8) 

Figure 32 

magnitude a harmonic force acting on the system will produce, approxi- 
mately, a harmonic motion. In other words, we expect motions due to 
spurious disturbances to die out quickly through the various dissipation 
mechanisms inherent in any real physical system, whereas a more or less 
periodic motion will persist through the action of the harmonic force 
applied. For a small Coulomb damping force this periodic motion can be 
approximated by a sinusoidal motion. Let the resulting motion be 
represented by 

x(t) = |A| cos (a> f t — a) 

The energy dissipation through Coulomb damping per cycle is evidently 
4/|^|. On the other hand, the energy dissipation per cycle for viscous 
damping is Trc^ 2 ^, according to (57). Hence we may obtain the equi- 
valent viscous damping coefficient by 

Af\X\ = *c\X\*<o, 


C = 

\0) i 


Substituting this equivalent damping factor into (41) or (47), we have 

|A| = 


Solving this equation for Ul, we have 




~ (*)' 

—moj f 2 + k 2 



■ (- l Y 

8, t 


In examining this equation we find that the damping parameter is now the 
ratio between the friction force and the amplitude of the applied force 
f/F . When this ratio reaches the value 77-/4, the equation can no longer be 
used, since it yields imaginary values. Within the range of f/F values for 
which the equation can be used the amplitude tends to infinity when the 
frequency of the applied force approaches the natural frequency of the 
system. The existence of a limiting value of f/F for the validity of the 
analysis and the occurrence of an infinitely large amplitude in a system 
with energy dissipation mechanism can be explained as follows. According 
to (56), the work done per cycle by the external force is nF \A\ sin a and the 
work dissipated by friction force is 4/UI. Since both are proportional to 
UL the system can achieve an energy balance only by adapting its motion 
to a proper phase lag a. 50 

sin a = — ^ 


If the friction / is so large that sin a has to be greater than unity, a 
sinusoidal response is completely out of the question, even as an approxi- 
mation. Experiments and exact analysis show that for large values of/the 
response motion has "stops"; that is, during a cycle the mass will come 

50 Here we are accepting a metaphysical principle, which is borne out in a number of 
physical laws. This principle in essence says that nature is clever but lazy. Within the 
freedom allowed her, she always finds the way in which she can exert herself the least 
in reacting to changes imposed on her. 




to a complete stop or stops before moving further. Such motions are 
qualitatively described in Fig. 33. The phase angle a defined by 

a = sin 


is multiple-valued. Let the principal values of the foregoing equation be 
a x and a 2 with 

a 2 > a x 


a-i 4- occ 



Analogous to the forced vibration of an undamped system, there is a 
sudden jump of the phase angle from ol 1 to a 2 as the frequency of the force 
passes through the natural frequency of the system. This jump discon- 
tinuity exists even in the exact solution, although there the phase angle 
also varies with frequency ratio elsewhere. In a real physical system the 


value of a must change rapidly, and near resonance condition it assumes a 
value between a x and a 2 , say a', Since 

sin a' > sin a x = sin a 2 

the energy input is larger than that which can be dissipated through 
Coulomb damping, and the amplitude grows excessively. 

The accuracy of this approximate analysis depends upon both the 
friction ratio f/F and the frequency ratio. Generally speaking, the accur- 
acy is good when the amplitude is large. This makes the analysis of 
practical value, since usually only the presence or the absence of large 
amplitudes is of engineering interest. As a rough guide, if the analysis 
yields a magnification factor \M/d st greater than 1.5, the error involved is 
only a few per cent. 

1.15 Theory of Galvanometer and Moving-Coil Instruments 

(a) differential equation of operation 

Moving-coil instruments of different designs, typified by a D'Arsonval 
galvanometer, are widely used for detecting and recording electrical 
signals from a variety of sensing elements, such as strain gages, thermo- 
couples, and vibration pickups. To a high degree of approximation such 
instruments behave like ideal linear vibratory systems having one degree of 

Basically a D'Arsonval galvanometer movement consists of a moving 
coil of moment of inertia/, a magnetic field that exerts a torque on the coil 
proportional to the current flowing in the coil, and an elastic spring that 
produces a restoring torque as the coil turns (See Fig. 80). The motion of 
the coil may be damped by immersing it in a viscous oil. In the more 
delicate types used in recording oscillographs the restoring torque is 
provided by the elasticity of the coil itself instead of a separate spring. 

A damping torque is also produced electrically in the following manner. 
When the coil turns with an angular velocity, it acts as the armature of an 
electrical generator and thereby generates a back electromotive force 
(emf for short). This emf causes a back current to flow if the galvanometer 
circuit is closed, and this back current produces a torque opposite but 
proportional to the angular velocity in the same way as a viscous damping 
torque. Since the back current is inversely proportional to the total 
resistance in the galvanometer circuit, so is the equivalent damping 
constant. If the circuit is open, the total resistance in the circuit is infinite, 
and the oscillation of the coil is therefore not damped electrically. [See 
also Art. 3.14(b).] 


The differential equation of motion of a D'Arsonval galvanometer 
without viscous damping is therefore 

^R + R g R + R g U 

6 = deflection angle 

J = moment of inertia of the coil 

k = spring constant of the meter movement 

C = deflecting torque per unit current 

e{t) = emf generated by the sensing device 

R g = resistance of the galvanometer coil 

R = resistance external to the galvanometer 

B = back emf per unit deflection velocity 51 

In practice, it is neither convenient nor necessary to describe the 
characteristics of a galvanometer by so many constants. The following 
four quantities are usually given in the specifications : 
S — galvanometer constant or sensitivity which is the galvanometer 

deflection per unit current when both are steady 
co = natural undamped (circular) frequency 
R c = critical damping resistance, that is, external resistance required to 

produce critical damping 
R g = resistance of the galvanometer coil 

The relationships between these two sets of constants are 

c c 2 k 

S = - or = - 

k J 

c c = iVjk (ii) 

Rc + *a 

By dividing the original equation by k and utilizing the foregoing relation- 
ships, we have 

co 2 co\R + Rj R + R ( 

e(t) (iii) 

Having written the equation in this form, we realize that the units of 
measurement of 6 can be arbitrary as long as the same unit is used in 
defining S. For a given set-up of instrumentation we can also lump 
together S/(R + R g ) as the deflection per unit signal strength and consider 
e as the signal strength. 

51 The constants B and C are equal. This fact is shown in Art. 3.14, but it is of no 
consequence here. 



In an ideal instrument the response d(t) should be proportional to the 
signal e{t). In an examination of the differential equation (iii) it is seen 
that such idealized characteristics can be approached if oj is high enough 
so that the first two terms in the equation are small. This means that the 
undamped oscillation of the galvanometer must be relatively fast in 
comparison with the rate of signal change, or 

\e\ > \e/(o\ \e\ > |e?/w 2 | 

However, in practice, this desirable condition is not easy to achieve 
without sacrificing the sensitivity of the meter. This is because a high 
value of co implies a stiff spring or a light coil ; a lighter coil means fewer 
turns, smaller coil loop, or finer wire, and finer wire results in higher R g . 
Any of these conditions result in less deflection per unit signal strength. 52 
Therefore, in choosing galvanometers for instrumentation a compromise 
has to be made between sensitivity and fidelity of response. 

Although the inertia and elastic stiffness of the galvanometer movement 
cannot be arbitrarily varied because of constructional practicability, 
the amount of damping is relatively easy to adjust; and, by a proper 
adjustment of damping force, the performance of the instrument can be 
much enhanced. First, suppose the galvanometer is to record steady-state 
sinusoidal signals of different frequencies. Evidently, it is desirable for 
the amplitude of the response to be proportional to that of the signal, 
regardless of its frequency. In other words, in the ideal situation the 
magnification factor should be independent of the frequency ratio. Figure 
16, however, shows that this is not exactly realizable in a simple spring- 
mass damper system. The most we can accomplish is an approximately 
constant amplitude of response for a limited frequency range. Such a 
condition exists for instruments with a moderate amount of damping at 
signal frequencies below the natural undamped frequency of the instru- 
ments. Figure 34 shows the magnified lower left portions of the curves 
in Fig. \6a. It is seen that some of the curves are reasonably "flat" within 
the frequency range shown. We noticed also that for high values of cjc c 
the point (0, 1) is a maximum and for low values of cjc c it is a minimum. 
As shown in Exercise 1.17, the transition takes place at cjc c = l/\ 7 2 when 
the point (0, 1) becomes a minimax, so that both the slope and the curva- 
ture are zero, and the curve has a broad plateau. This value of damping is 

52 A way of compensating the loss sensitivity in using a lighter coil and stiffer suspen- 
sion is to increase the magnetic-field strength of the pole pieces. This usually results in 
bulkier instruments. Compactness is of importance in such instruments as oscillo- 
graphs, which contain scores of galvanometers in a small enclosure. 




then a theoretical optimum. In practice, a lower damping ratio is usually 
chosen because, by allowing the curve first to go up a little and then to 
come down, the usable frequency range is further extended, although it is 
not so flat at the beginning as the curve corresponding to the theoretical 
optimum damping of 0.707. For instance, with 65 per cent of critical 




V -( 


C K 

= 0.65\ 


N c / 

c r = ( 




Sc/c c 

= 0.75 N 


c/c c - 0.80\ 
1 1 \ 


0.10 0.20 0.30 0.40 0.50 

Frequency ratio ooJcc 

Figure 34 



damping, the variation in frequency response for signal frequencies up to 
60 per cent of the natural frequency of the instrument is within ±1.3 per 

It must be emphasized that even within the frequency range, where the 
frequency response is relatively constant, the phase difference between the 
signal and the response varies appreciably. Hence, if the signal contains 
harmonics of different frequencies, their phase differences will not be 
preserved in the response. This results in a distortion of the waveform 
called a. phase shift. This type of distortion may or may not be important 
in a given application of the instrument. 

The criterion for optimum damping can also be based upon the response 
of the instrument to step signals. This is especially appropriate if the 
instrument is to record transient signals faithfully. Let S be the sensitivity 
of the instrument. The response of an ideal instrument to a step function 
S(t) is then SS(t), or simply S for t > 0. 53 

53 We assume here that the signal is in the form of a current. If it should be a step 
voltage, replace S by SI(R + R„). In any event, by sensitivity or meter constant one 
means the response per unit signal strength when both are constant with respect to time. 


The response of the actual instrument is u(t), which is given by (696) 
with \\k replaced by S. We can then define the error E{t) as 

m . , - f 

It can be verified that 

E{t) = 

— £?-< tan W^ cos (gU - (5)1 

-0) r . J 

The time average of the error in a given time interval r is 

\[ mdt 

and the mean square error is 

i f T {£(r)} a A 


Other things being equal, these integrals are a function of c/c c ratio. The 
criterion for optimum damping may then be the minimization of these 
integrals in a specific interval. 

With a moderate amount of damping the error function E(t) becomes 
quite small after t reaches a value equal to a few times the natural 
period of the instrument, so it is quite reasonable to optimize the amount 
of damping by minimizing either of the following two integrals: 

I* oo C oc 

\E\dt and E 2 dt 

Jo Jo 

Both of these improper integrals are convergent. 

The first integral cannot be evaluated conveniently. 54 Since this cri- 
terion is somewhat arbitrary and since the results, if carried out, cannot 
differ much from those by minimizing the second integral, we choose the 
second criterion instead. This leads to an optimum damping ratio of 
cjc c = \. The details of this procedure are left to the readers as an exercise. 

Another and perhaps commoner procedure of choosing an appropriate 
damping ratio is by minimizing the so-called response time, that is, the 

54 In case one wishes to evaluate it, the integral can be expressed as an infinite series 
of definite integrals, each covering an interval of time in which the cosine function does 
not change sign. Except for the first term, the values of the succeeding integrals form a 
geometrical progression with a common ratio of eA/ 2 . The sum of the series can thus be 


time required for the instrument to settle down into an oscillation between 
prescribed limits. This procedure usually amounts to the determination 
of the minimum damping ratio, so that the maximum "overshoot" is less 
than the amount prescribed, although the two criteria are based on two 
slightly different physical considerations. 55 For example, with a prescribed 
error of ±5 per cent, the shortest response time is 43 per cent of the period 
of natural oscillation, and it corresponds to a damping ratio of 0.69, which 
is also the value of the minimum damping ratio for a maximum overshoot 
of 5 per cent. This overshoot takes place at t — 0.65T. 

The different considerations involved in selecting an optimum damping 
ratio, as discussed, lead to the observation that for a general purpose 
instrument a damping ratio of 0.65 is a good compromise. 

In concluding this article, let us remark that although a moving coil 
galvanometer was chosen for this study many of the results, observations, 
and methods of analysis are applicable to other types of instruments and 
control equipment. 

1.16 Seismic Instruments and Transducers 

The type of device discussed in this article is essentially a single-degree- 
freedom vibratory system mounted on a frame or in an enclosure which 
can be attached to a moving body whose motion is to be measured or 
converted into an output signal. The total mass of the system and its 
mounting must be relatively small so that it will not affect the motion of 
the body to which it is attached. 

(a) seismograph 

The oldest instrument of this type is the seismograph, an instrument 
for detecting the motion of the earth's surface. In its simplest form it 
consists of a spring-mass system of very low natural frequency enclosed 
in a frame attached to the ground, as schematically shown in Fig. 35. 56 
The input to the system is the motion of the ground (hence of the frame), 
and the output is the relative displacement between the frame and the mass. 
This relative displacement can be recorded by attaching a tracing pen to 
the mass and attaching graph paper to the frame. 

Let x be the displacement of m, x s , the displacement of the frame, and 

68 See Draper, Mckay, and Lees, Instrument Engineering, Vol. II, McGraw-Hill, 
New York, 1953, pp. 264-265. 

56 We assume in Fig. 35 that the motions are in the vertical direction. The mass 
could be suspended to have freedom in three dimensions. However, since the three 
motions are independent of one another, the principles involved are essentially those of 
a single-degree-freedom system. 




x r , the relative displacement x s — x. The differential equation of motion is 

mx + k(x — x s ) = 

mx r + kx r = mx s 

x r + oj 2 x r == £ s 

From this equation it can be inferred that if the natural frequency of the 
system is very low, relative to the two accelerations, then 

x\ == x\ 


Figure 35 

This implies that the response is equal to the signal when everything starts 
from a rest condition at t = 0. To achieve this approximate relationship 
physically, we need to have a very heavy mass 57 suspended by very soft 
springs. As the frame moves because of earthquake or underground 
explosion, the heavy mass stands almost still, and any rapid vibration of 
the ground is reflected directly by the relative motion between the mass and 
the frame. The principle of a seismograph is similar to that of certain 
vibration pickups. 

(b) vibration pickups 

There are several types of vibration pickups whose construction is 
essentially a spring-mass system (with or without a damper), mounted in 
a housing which can be attached to a vibrating machine or structure, as 
shown in Fig. 36. We give them the generic name seismic instruments 
because they are related to seismographs in their operational principle. 

57 Even a very heavy mass is still not heavy enough to affect the motion of the earth's 




The characteristic feature of a seismic instrument is that its output or 
response is generated by the relative motion between its housing and the 
mass inside. This output may be in the form of a mechanical movement, a 

tr 1 -^ 

*~x s 

Vibrating structure 
Figure 36 



— . i 


o *■ 

8 Output 

o *- 



Moving coil pickup 




.crystal or other 
sensitive device 

Variable reluctance pickup 






Figure 37 

moving light beam, or an electrical voltage. Figure 37 shows schematically 
some typical instruments whose output is in electrical form. 


The strength of the output may depend upon the relative displacement 
or the relative velocity between the housing and the mass, and accordingly 
the pickup is said to be displacement-sensitive or velocity-sensitive. The 
response may be intended as a measurement of the displacement, the 
velocity, or the acceleration in the motion being measured, and accordingly 
the instrument is called a vibrometer, a velocity pickup, or an accelerometer. 
Regardless of what we consider as the input signal and what the pickup 
gives as response, the basic differential equation for a seismic instrument is 

mx + c(x — x s ) + k(x — x s ) = 

in which x is the displacement of m and x s is the displacement of the 
motion to be sensed by the pickup. Let 

and we have 

mx r -f cx r -f kx r = mx s (i) 

If the instrument is a vibrometer and it is displacement-sensitive, as the 
seismograph previously described, (i) is the relationship between the 
signal and response, and the frequency response of the vibrometer is 
described by 


w- (i . 

\V Vib 

1 - 

mojf + k + icco f 
k + icco f 

k — mco f 2 + icto f 
= 1 - T r (ko f ) (ii) 

in which A s and X r are the complex amplitudes of the signal and the res- 
ponse, respectively. 

According to (80), the function T r is the transfer function for the trans- 
missibility of the vibratory system inside the seismic instrument. With 
the help of Fig. 22, we may conclude the following: 

(1) Since T r approaches zero as co f becomes large, the response is 
nearly constant if the natural frequency of the instrument is sufficiently 
below that of the vibration being picked up. On the other hand, the 
instrument is insensitive to slow changes in the signal. A signal such as 
the one shown in Fig. 38a is picked up by the instrument as one represented 
by Fig. 386. This can be an advantage or a disadvantage, depending upon 
applications. Note that the situation here is exactly the opposite of 
galvanometer response. 

(2) Since T r approaches zero faster with less damping, there is no 




advantage in introducing any damping in the system. Therefore, a 
vibrometer is designed with a minimum amount of damping. 

Let us now take the case of a velocity pickup whose output is propor- 
tional to the relative velocity of the motion inside. The signal is then taken 
to be v s = x s and the response to be v r = x r . Evidently, by differentiating 
(i) with respect to /, we have 

mv r + cv r + kv r = mv s 

So if this time we let our signal k 8 e la>ft be v s and the response X r e l<0 f l , 



Figure 38 

v r , the relation between X r and X s would be the same as before, and all the 
observations we made about a vibrometer would hold true for a velocity- 
sensitive velocity pickup. 

Now let us suppose that the instrument is an accelerometer and its 
output is proportional to the relative displacement. The input signal is 

and the response is 

l s e % ^ 1 = x s 

;./<v = Xr 

According to (i), the relationship between X r and X s is then 
( — mcQ f 2 + ico) f + k)X r = mX s 

T a (ia) 



k — mco f 2 + icoo f 

co f 2 2c co f 

1 --! + *- — 

or c c oj 



The denominator in this expression is the same as that of (46), and the 
numerator is a constant of the instrument. The response of an accelerom- 
eter is therefore proportional to its magnification factor, and the 
situation is the same as a D'Arsonval galvanometer. To repeat what we 
said about a galvanometer: 

(1) A displacement sensitive accelerometer should have a natural 
frequency that is high in comparison with that of the signal. On the other 
hand, by making its natural frequency high, one sacrifices its sensitivity, 
which, according to (iii), is proportional to 1/w 2 . 

(2) For best results an accelerometer has to be damped to a c\c c value 
between 0.6 and 0.7. The damping in an accelerometer serves two pur- 
poses. The first is to extend the usable frequency range of an instrument 
of a given sensitivity. The second is to prevent resonance with the higher 
harmonics of the input signal. For instance, if the accelerometer has a 
natural frequency of 120 cycles per second and the signal is a slightly 
distorted sinusoidal motion of 40 cycles per second, the instrument 
without damping would be resonant with the third harmonic of the signal 
and give entirely erroneous readings. Note that in a vibrometer the same 
situation will not be obtained if the natural frequency of the instrument is 
below that of the fundamental frequency of the signal. 

Another remark we made about the phase shift in a galvanometer with 
damping also applies here; that is, by extending the frequency range of 
the instrument through proper damping, we do not eliminate the phase- 
shift distortion, which is important only when the waveform of the signal 
is of interest. 


Most modern instruments give their outputs in the form of electrical 
signals. These have to be converted by external means into the motions 
of a pointer or recording pen. The conversion device is in itself a signal- 
transferring system. It is then possible to introduce an electrical network 
into this device, which has a signal-transfer characteristic compensating in 
part for the inaccuracies in the sensing instrument. For instance, if an 
accelerometer is damped to 70 per cent of the critical, its response falls off 
for signal frequencies above 20 per cent of its natural frequency. By 
introducing a network whose response shows a proper rise above the same 
frequency, we can extend the usable frequency range of the combination. 
Another familiar example is a magnetic phonograph pickup, which is 
essentially a velocity-sensitive seismic instrument in which the arm is 
equivalent to a seismic mass and the needle is equivalent to a mounting 
frame. Since most records are cut to be played by displacement sensitive 




pickups, the use of magnetic pickups requires compensation in the play- 
back amplifier. 

The point to be emphasized is that with the available flexibility in 
compensating network design a flat frequency response of a sensing device 
may not always be of the utmost importance as long as its response 
characteristics are known. 

1.17 Vehicle Suspension 

The suspension system of a vehicle may be analyzed in the first approxi- 
mation as a vibratory system having a single degree of freedom. As shown 



1 ^ 


->- y = vt 

Figure 39 

iii Fig. 39, the system is equivalent to a seismic instrument, and its differ- 
ential equation of motion is 


mx r + c(x r — x s ) + k(x r — x s ) = 
mx r + cx r -f kx r = cx s + kx s 


where x s and x r are the vertical displacements of the wheel and the vehicle, 
respectively. If the profile of the road surface is given by x s =f(y) and the 
velocity of the vehicle is v, then 

x s =f(vt) and x s = vf'(vt) (iii) 

So the left-hand side of (ii) is a function of /. When the road surface 
irregularity is more or less periodic, the frequency response of the system 
is of immediate interest. It must be pointed out that although the differen- 
tial equation is the same as that for a seismic instrument the response of 




the system is taken to be the motion of the vehicle body rather than the 
relative displacement. The frequency response is then 

(-) = 

k + icoOf 

k — mo) f 2 + ico). 

= T r (i(o f ) 

which is the transmissibility of a spring-mass-damper system. The 
frequency co f is determined by the profile of pavement irregularities as 
well as the speed of the vehicle v, A look at Fig. 22 enables us to make 
the following observations : 



x s =J~(t) 

Figure 40 

(1) Insofar as vibration absorption is concerned, it is desirable to use 
soft suspension springs to lower the natural frequency of the system. 
Very soft suspension springs, of course, have problems of their own. 
These include lateral instability and excessive changes in road clearance 
and vehicle height between loaded and unloaded conditions. 

(2) At low speeds, which correspond to frequency ratios below 1/V2, 
damping is necessary to cut down the vibration transmitted from the wheel 
to the body, whereas at higher speeds the existence of damping is actually 
disadvantageous insofar as steady-state road disturbances are concerned. 

An important consideration in the design of vehicle suspension concerns 
its ability to absorb bumps and holes in the pavement. This ability can 
best be assessed by its indicial response. By definition, the indicial response 
in this case is the solution to (ii) for x s (t) — S(t) with x r (0) = 0. There is, 


however, a mathematical difficulty in solving this problem by the classical 
method, since with x s being a step function cx s is infinite at t = 0, although 
it is zero at all other times. There are other methods, including the use of 
(96), whereby this difficulty can be removed. But we have dealt with this 
problem before in Art. 1.11, in which it is shown that the differential 
equation (78) is equivalent to a pair of equations (77). Borrowing the 
subsequent results, together with {69b), we have 

u(t) = ci— e - {cl2m)t sin co c t) + ( 1 - — e~ {cl2m)t cos (co c t - 6)) 
\mco c I \ co c / 

With the help of (27), this expression can be simplified to yield 


u r {i) = 



co c 

e -(c/2m)t 


(co c t + 6) 

From this 



u r \t) 


CO 2 

— e 
co c 

-(c/2m)« ^ 

I (co 

J + 26) 


Now we see that at t = the foregoing expression does not vanish, since 

,,^ 0j2 • ~ c 2coc c 

u r '(0) = — sin 2d = = - 

co c c c m 

Hence this does not satisfy the initial condition x r (0) = 0. However, 
when we examine the physical meaning of indicial response, we see that 
the system is assumed to have zero displacement and zero velocity for all 
t < 0. They are zero at / = only if they are both continuous at t = 0. 
Such continuity exists if the signal-response relationship is of the form (97); 
but, when the relationship is of the form (98), the derivatives of the 
indicial response may not be continuous. Returning to the problem on 
hand, we see that when the wheel is suddenly given a displacement the 
body of the vehicle will suddenly acquire a velocity of c/m. This statement 
is true also for c\c c > 1 , since (iv) and (v) are true if co c and 6 are allowed to 
be complex. 

This sudden velocity change 58 of the vehicle body represents a shock or 
impact which should be kept small. This means that the value for c/c c 
should not be too large. On the other hand, to prevent large oscillation, 
c\c c should not be too small. A good compromise is for c\c c = 1. 

68 In reality, the tire and wheel structure is not absolutely rigid so that the acceleration 
will not be infinite. 


1.18 Structural Damping and the Concept of Complex Stiffness 


In a real vibratory system the restoring force of the "spring" is not 
entirely conservative. Thst is to say, cyclic deformation of the spring 
consumes mechanical energy. The dissipative forces involved originate 
either in the interior of the spring material or at regions between two pans 
of a built-up spring. The outward manifestation of such dissipative forces 
is described by the general term structural damping. As a rule, forces due 

4 Stress 


Figure 41 

to structural damping are small, but often their presence affects the dynamic 
behavior of a vibratory system, especially if the vibration is self-excited. 59 

In this article we discuss a simple form of structural damping and its 
effects on the vibratory motion of a single-degree-freedom system. 

It is generally accepted that when structural damping is caused by the 
material of the spring in a spring-mass system the hysteresis of the spring 
material under cyclic deformation is responsible for the energy dissipation. 
Hence we have the term hysteresis or hysteretical damping. Precise measure- 
ments on the stress-strain relationship of most real materials show that 
even at stress levels much below their accepted elastic limits cyclic straining 
produces hysteresis loops such as that shown in Fig. 41 . The area enclosed 
by the loop represents the mechanical energy dissipated by the material 
during one complete stress-strain cycle. 

59 A vibration is said to be self-excited when the external force becomes a periodic 
excitation by virtue of the vibration itself. For instance, the stroking of a bow on a 
string produces a periodic force on the string as the latter vibrates. Also, the flutter of 
an airfoil is excited by aerodynamic forces which become periodic as the airfoil flutters. 



Using a rotating beam loaded by a dead weight, Kimball and Lovell 60 
showed that for most materials the energy dissipated per cycle is indepen- 
dent of frequency over a wide frequency range and proportional to the 
square of the stress (or strain) amplitude of the cycle. This observation 
can be translated to mean that the hysteresis loop is not affected by the 
rate of straining and that a change in the stress amplitude produces a 
change in the size of the loop but not its shape. More modern measure- 
ments by Lazan 61 indicate that the foregoing statement is approximately 
true for many materials in a limited range of frequencies and stress 
amplitudes. The actual phenomenon, of course, is exceedingly complex 
and is dependent upon a host of mechanical and metallurgical factors. 


When a spring-mass system with a small amount of structural damping 
is excited by an external harmonic force, it is reasonable to expect that the 
resulting steady-state vibration will be a harmonic oscillation with the 
frequency of the excitation co f . Let the amplitude of this oscillation be 
|A|. The law of structural damping, according to Kimball and Lovell, 
then states that the energy dissipation E d per cycle is 

E d oc |f (i) 

In the meantime, according to (57), the energy dissipation per cycle by 
viscous damping under the same circumstance is 

E d = 7TCCO f |A| 2 

Therefore, a system with structural damping in steady-state vibration 
may be analyzed as a viscously damped system having a damping constant 
c inversely proportional to the frequency. In other words, the linear 

mx -\ x + kx = F cos cu f t (ii) 

CO f 

has a steady-state harmonic solution in which the energy dissipation per 
cycle is independent of the frequency co f although proportional to the 
square of the amplitude. The constant h is determined by the property of 

60 "Internal Friction in Solids" by A. I. Kimball and D. E. Lovell, Physical Review, 
Vol. 30, December 1927. 

61 "A Study with New Equipment of the Effects of Fatigue Stress on Damping 
Capacity and Elasticity of Mild Steel" by B. J. Lazan, Trans. Am. Soc. Metals, Vol. 4, 




£ 2.0 





85 60 



1.0 1.5 2.0 

Frequency ratio co^/w 



S^ ^^-^^"^ 

/ / X* 


i i 


1.0 1.5 2.0 

Frequency ratio co^/w 




Figure 42 


the spring-material, and it has the dimension of k. Often one writes 

h = k?] 

in which r\ is a nondimensional quantity called the structural damping 
coefficient and is usually less than unity. The steady-state solution for 
(ii) can thus be obtained by replacing c with hjco f in (41) of Art. 1.6. The 
result is 

x = Ul cos (co f t — a) 

h ri 

tan a 

—tncOf 2 + k 

- o 

^0 d,t 

\/(-mco * + kf + h 2 I a)*\* , (hi) 

+ rf 

(■ - w 

Plots of the magnification ratio \k\/6 st and the phase lag a versus the 
frequency ratio co f /co for different value of r\ are shown in Fig. 42. It is 
instructive to compare these plots with Fig. 16, which is designed for a 
viscously damped system. An interesting feature of Figure 42a is that the 
peaks of all the curves are at co f lco = 1 . 


Since (ii) is a linear differential equation having a solution of the form 
(iii), we can use a complex number representation 

x = Xe^f 1 (iv) 

by rewriting the differential equation of motion as 

mx -\ x + kx = F e H ' J ^ (v) 

Upon differentiating (iv) and putting into (v), we obtain 

x = icOfXe^f 1 = ico f x 

mx + (ih + k)x = F e ia>ft 

mx + k(\ + irj)x = F e' CO}t (vi) 

Writing the equation of motion in the form of (vi) has the advantage of 


simplicity as well as the disappearance of co f from the left-hand side of the 
equation. The complex quantity 

K = k + ih = k{\ + irj) 

is called the complex stiffness? 2 This complex quantity represents the 
elastic and the structural damping forces of the spring at the same time. 
Such representation is particularly advantageous in dealing with systems 
having many degrees of freedom. The vectorial representation of (vi) is 
shown in Fig. 43. 

Figure 43 

(d) vibration of structurally damped system 
in the absence of external excitation 

It must be emphasized that all the results presented are valid only when 
the vibration is known to be a harmonic oscillation. It is obvious from (ii) 
that in the absence of external excitation the homogeneous differential 

mx -\ x + kx = 

(J0 f 

has no physical meaning, since it includes an undefined quantity co f . 
Unfortunately, co f does not appear at the left-hand side of (vi), and this 
fact led some investigators into trying to solve the problem by solving the 

mx + k{\ + irj)x = (vii) 

We must realize that (v) can be replaced by (vi) only because we know the 
solution to be of the form (iv). 63 

62 As a matter choice, some authors call it the complex damping. 

63 See the discussion following (49) in Art. 1.7. Also, for a physical problem meaning- 
fully described by (vii), see Art. 2.14(c). 


The experimental evidence presented in (a) was from a steady-state 
phenomenon. We really have no adequate physical data to analyze a 
damped oscillation. It is, however, reasonable to believe that if the 
structural damping force is small, so that the decay of the resulting damped 
oscillation is gradual, the motion may be assumed to be the same as that of 
a viscously damped system with an equivalent damping constant given by 

_ h 
oj c 
The equation to be solved is then 

mx -\ x + kx = (viii) 

co c 

where co c is the damped frequency of the vibration. This approach was 
originally suggested by Collar. 64 The solution to (viii) can be obtained by 
replacing c with hjo) f in (19) and (19a). 

x{t) = Ce- {htl2m ^ cos (oo c t - a) (ix) 

The damped frequency co e is to be determined by (20) 

V 4mkoj 2 - h 2 

«r = = 

2moj c 
Solving for co c , we obtain 

k + Vk 2 -h 2 /l + VI — >7 2 
M < = J 2Tn = °V 2 (X) 

The oscillatory solution (ix), of course, can be valid only for r\ < 1. An 
aperiodic solution for r\ > 1 has no physical meaning because we know 
nothing of the law of structural damping for such motion. It appears that 
according to (x) the limiting value of co c for oscillatory motion is co/Vl 
instead of zero. This is not so mathematically. There is a second solution 
to (ix) which corresponds to the use of minus signs instead of the plus 
signs under the radicals in (x). The branch point corresponds to a loga- 
rithmic decrement of 277, a decaying rate too rapid to justify the extrapo- 
lation of Kimball-Lovell's law, as stated. 

A more lucid way of expressing the relationship between co c and r\ is by 
the use of the angle d defined in (27). 

co, / 1 + cos 2d 

— = cos d = 


64 See "The Treatment of Damping Forces in Vibration Theory" by R. E. D. Bishop, 
Journal of Royal Aeronautical Society, Vol. 59, No. 539, November 1955. 


In comparing this expression with (x), evidently 

rj = sin 2d (xi) 

(e) complex number representation of damped oscillation 
In complex number representation the solution to (viii) is in the form of 

x = le ,at 
where the complex frequency a is defined in (22) as 


a = co c + i — co — oj c (1 + i tan b) 


x = iax = io) c (\ + i tan b)x 

By substituting this into (viii), we obtain 

mx -f ih{\ + / tan b)x + kx = 

ma; + k{\ — r\ tan b + z^)x = 
Because of (xi) 

1 — 77 tan b = 1 — 2 sin 2 (5 = cos 2(3 = Vl — ?y 2 

We have, finally, 

mx + &(Vl - ?f + />?> = 

mx + ke 2ld x = 

Z7zi; + /c^ = (xii) 

Therefore, in this case the complex stiffness is the quantity 

Kl = k(V\ - yf + irj) = ke m 

Here the structural damping coefficient ?] enters not only into the imaginary 
part but also into the real part of the complex coefficient. This came about 
because of our scheme of representation and not because structural 
damping force has any physical effects on the conservative force of the 
spring. This observation is clear, as we examine the original differential 
equation (viii). 

We see, therefore, that for a given system with structural damping the 
complex stiffness k in the case of steady-state forced vibration is different 
from that in the case of a damped oscillation k ± . 65 For small v\ the differ- 
ence is, of course, small. It arises out of the difference in the structural 

65 For another complex number representation of the same hysteresis characteristic 
see Art. 2.14(c). 


damping laws used to analyze the two types of motion. The use of k for k 1 
in (xii) leads to (vii), which gives inadmissible results. 66 Myklestad, 67 by 
a different method of derivation, however, arrived at an equation for 
steady-state vibrations in which k x appears instead of k. 

mx + ke 2id x = F e i<a f f (xiii) 

He stated that certain features of his solution agree better with experi- 
mental observations, but there is some question 68 about the physical 
reasoning underlying his method of derivation. As a practical matter, 
since the structural damping force is generally small, the numerical results 
obtained from (viii) and (xiii) have no significant difference. 

(f) other types of structural damping 

The structural damping law of Kimball and Lovell has only limited 
validity. It has the advantage of yielding a linear differential equation. 
Therefore, whenever reasonable, this law is used for analysis. There are 
many actual systems with structural damping whose oscillations cannot 
be described by linear equations without gross error. For example, for a 
certain built-up beam Pian and Hallowell 69 found that hysteresis loss 
caused by slipping between its parts is approximately proportional to the 
third power of deformation while it is independent of the stress amplitude. 
Naturally, the analysis of such a system is more complicated. Since 
structural damping force can originate from a number of different physical 
phenomena, no unifying treatment of such forces is possible. 


1.1. Write the differential equations of motion for small oscillations of the 
systems shown. 

66 For instance, the solution to (vii) is oscillatory for all values of rj and co c increases 
with ?/. The solution is given in Art. 2.14(c). 

67 "The Concept of Complex Damping", N. O. Myklestad, Journal of Applied 
Mechanics, Vol. 19, No. 3 (September 1952) 

68 See "Concept of Complex Stiffness Applied to Problems of Oscillations with 
Viscous and Hysteretic Damping" by S. Neumark, Royal Aircraft Establishment, 
Report No. Aero. 2592 (V.D.C. No. 533.6.013.42) September 1957. 

69 "Structual Damping in a Simple Built-Up Beam", by T. H. H. Pian and F. C. 
Hallowell, Proceedings of First U.S. National Congress of Applied Mechanics, pp. 
97-102, 1951. See also "Structural Damping" by T. H. H. Pian, Chapter 5 of Random 
Vibration, edited by S. H. Crandall, MIT Technology Press, Cambridge, Massachusetts, 







O m 


Exercise 1.1 


1.2. A torsion-bar suspension system for vehicles may be approximately 
represented by a weight hanging on an L-shaped round bar as shown. Find the 
natural frequency of the system in terms of a, b, d, E, G, and W. 

E = Young's modulus G = Modulus of rigidity 

Exercise 1.2 




1.3. To find the moment of inertia of a flywheel, a certain engineer hangs the 
wheel on a nail and measures the period of swing. The wheel weighs 60 lb, the 
point of suspension is 10 in. from the center of the wheel. If the period of the 
swing is 1.45 sec, what is the moment of inertia of the wheel about its centroidal 
axis? The wheel is assumed to be well balanced. 

1.4. A spring-loaded cam follower follows an eccentric radius r and eccentricity 
e. The mass of the follower is m, and the spring has a spring constant k and a 
natural length that is longer than its maximum length in position by an amount 
equal to 2e. 

(a) Show that for small e\r the follower has a simple harmonic motion when 
the eccentric rotates with uniform angular speed oj f . 

(b) What is the maximum allowable oj f so that the follower will not leave the 

Exercise 1.4 

1.5. Show that 

A cos (cot — a) + B cos (cot — /?) = C cos (cot — y) 

where A, B, and C are three sides of a triangle with the respective opposite angles 
being ±(y - p), ±(<x - y), and xr ± (a - p). 

1.6. A system consists of four uniform identical rigid bars each of length a 
and mass m, linked together by frictionless pin joints and a linear spring of 
spring constant k and natural length a. Find the natural frequency of the system 
by Rayleigh's principle. Assume that the mechanism lies on a smooth horizontal 
table so that its center of mass is stationary during oscillation. 



Exercise 1.6 

1.7. Draw the phase trajectory of a simple pendulum of length L swinging a 
total angle of 180°. Compare its shape with that of an ellipse having the same 
major and minor axes. 

1.8. A plumb bob of effective length / is hung on a nail on a vertical wall. 
Directly under the nail of suspension is another nail which catches the string 
when the plum bob swings to one side but not when it swings to the other side. 
If the distance between the nails is //4, what is the period for small angle swing 
and what is the ratio of the two maximum angles the string makes with the 
vertical ? 

Exercise 1.8 

1.9. A piston of mass m "floats" in a cylinder of height h and cross-sectional 
area A ; the cylinder contains a perfect gas. Assuming that the compression and 
rarefication of the gas are reversible adiabatic as the piston is set into small 
up-and-down oscillation, find an expression for the frequency of such oscillation. 













Exercise 1.9 

1.10. A small ring slides back and forth without friction along a wire bent 
to a curve in a vertical plane described by the equation 

x 1 x* 
2a a A 

Find the period of its oscillation when x max is 0.2a. 


T = \m 


= hn(\ + y' 2 )x* 

V = mg(y - y m ) T + V 
d(T + V\ 

( y - Vrr 

d_ y 



For small 

+ g(y' - y' z + 2y m y'y" - 4y m y' 3 y" - 2yy'y") = o 

Neglect all terms higher than third order of x. 

(a) The solution expressed by (15) can also be written as 

x{t) = e-^ 2m ^(A cosh wt + B sinh wt) 

in which w = V(c 2 /2m) — 4mk = ia> c . By the usual method, determine 
the constants A and B in terms of x and x . 
(b) Utilize the formula 

cos id = cosh 6 


sin id = i sinh 6 

to show that (19) and (23) will lead to the same answer as (a). 

(c) Show that, as in Fig. 7, a hyperbola may be used to relate cjc c and w/a>. 
If we let cjc c = cosh € and wjco = sinh <?, what is the geometrical meaning 
of e? 

(d) Show that (18) can be obtained by letting co c approach to zero in (19) 
or by letting s x = to and s 2 approach s ± . 


1.12. Show that the angle 6 in Fig. 8 is the 6 in (27). 

1.13. Show that 

In — — = A 

x(t + T) 

1.14. Draw the phase trajectory of a damped single-degree-freedom system 
with c\c c = 0.5 and compare its shape with Fig. 8. 

1.15. A suspension galvanometer has the following periods of swing: open 
circuit 3.00 sec; 2000 ohms across the terminals, 3.20 sec; 1000 ohms across the 
terminals, 3.60 sec. What is the coil resistance and the damping resistance of the 
galvanometer? Is this a practical way to determine the two resistances of a 
galvanometer experimentally? Why? 

1.16. Plot two cycles of the envelope C in (36) for B = 0.7 A. 

1.17. In Fig. 15, if a circle is drawn with its center at the origin and tangent 
internally to the parabola, the radius of the circle and the point of tangency 
correspond to the maximum frequency response and its phase lag for a given 
damping ratio. Using methods in analytical geometry, show that 

(a) The radius of the tangent circle is sin 26, hence the maximum frequency 
response is 

A-ml^st = CSC 2^ 

(b) The corresponding phase lag a m is 

a m = tan _1 ( Vcos 26 esc <5) 

(c) The frequency response curve has a peak only when 

6 <- or -<1/V2 

4 c c 

1.18. Improve the accuracy of the period obtained by Rayleigh's method for 
the system shown in Fig. 28, when the 6 has an amplitude of 30 : and (gja) = 

1.19. A spherical ball of weight w and radius r rolls back and forth without 
slipping in a plane motion inside a bowl whose surface is a surface of revolution 
formed by the sinusoidal curve 

y=r\\ -cos^) 

Find the Rayleigh's method its period for small oscillations. 

1.20. An overdamped single-degree-freedom system is set into motion with 
initial displacement x and initial velocity ^ . Show that the condition for .v(t) > 
at all t < oo is that x > s 2 »V 

1.21. A damped single-degree-freedom system has zero initial velocity. Show 
that, other things being equal, the integral 



is a minimum for c = 0.5c c . 
Hint. Use (15) and (17) for your analysis. 


1.22. The signal from a 4-arm Wheatstone strain gage bridge is to be recorded 
by an oscillograph using a suspension coil galvanometer which has the following 
specifications: galvanometer coil resistance 60 ohms, critical damping resistance 
140 ohms, and natural frequency 120 cycles. If the gage resistance is 360 ohms 
and the record shows a steady-state sinusoidal signal of 90 cycles per sec, what 
is the percentage error in the record? Assume that the bridge and the oscillo- 
graph are properly calibrated by static tests. 

1.23. Let the signal input to a galvanometer be 

x s (t) =0 t < 

x s (t) = Kt t > 
in which A' is a constant. 

(a) Find the expression for galvanometer response x r (t), assuming that the 
galvanometer is calibrated for static readings. 

(b) Show that the error x r — x s approaches a constant value. 

(c) What would youestimate to be the optimumdampingratio in this galvanom- 
eter operation? 

1.24. The indicial response function u(f) may be represented by 

a n u(t) = 1 - e(t) 
where e(t) satisfies 

1(e) =0 t > 

ande(0) = 1, e'(0) = • • etc. 

(a) Use this approach to derive (69b) by the use of (19) and (23). 

(b) Use the same approach to find u(t) of a spring-mass-damper system for 
cjc c > 1 and c/c c = 1. 

(c) Obtain the answer for (b) by letting co c = iw for cjc c > 1 and by letting 
co c approach zero for c/c c = 1 . 

1.25. Draw the phase trajectory of the system in Fig. 27 with the initial 

x(0) = x(0) = x 

1.26. A conservative system consisting of n mass particles m lt m 2 , . . . , m n is 
constrained to move with a single degree of freedom so that the spatial coordi- 
nates of the particles are difTerentiable functions of a single configuration variable 
6, that is, x t = x0) Vi = Vi (d) z t = 2,(0); i = 1, 2, 3, . . . , n. 

Show that the kinetic energy of the system is given by 

T = \m(d)() 2 

in which m is a function of 6 only. 

1.27. A suspension coil galvanometer, whose movement constitutes a spring- 
mass system, has an undamped natural period of 3.6 sec and a sensitivity of 
1 in. deflection per microvolt. If this galvanometer is used in a circuit having a 
resistance equivalent to 64 per cent of the critical damping, and an emf of 1 
microvolt suddenly appears across the galvanometer terminal, find 

(a) the time required for the galvanometer to deflect 1 in., 

(b) the maximum overshoot, 

(c) the time-deflection curve for the first 10 sec. 



1.28. The idealized vehicle in Fig. 39 hits a bump on a pavement which is 
otherwise level and smooth. Let the profile of the bump be represented by one 
period of a cosine wave: 

/ 2ny\ L L 

x s = a\ 1 + cos— -I ~2 <2/< 2 

x s = elsewhere. 

Assume that a and R are small in comparison with L and that c/c c = 1. 

(a) Find the vertical motion of the vehicle. 

(b) What is the maximum horizontal velocity u the vehicle may have in passing 
over the bump without the wheel leaving the pavement? 

1.29. A platform of weight w is supported by a spring k and a shock absorber 
c. At / = the system is at rest and a weight W is suddenly placed on the 

(a) At what position will the platform finally come to rest again? 

(b) Using this position as the new equilibrium position of the system and the 
displacement from this position as the variable of configuration x, analyze 
the motion of the system. 


y 2 k 



Exercise 1.29 

1.30. If the weight W in Exercise 1-29 is dropped onto the platform from a 
height h, instead of being placed on the platform, what changes will you have to 
make in the analysis? (Assume that the weight does not bounce on impact.) 
What is the maximum force transmitted to the foundation in terms of W and h ? 
(Assume cjc c = 0.5.) 

1.31. What is the damping ratio the system in Exericse 1-29 should have if 
the requirement is that 

\x\ <0.02(W + w)lk 

for t > 3 T where T = 2irVkgl(W + w). 
1.32. Show that 

F{r)h{t -r)dr = F{t - r)h 




1.33. Find the transfer functions between x r and x s for the systems shown. 









x s = force on m, F 

x r = bending stress at A 


Filled length = L 
Fluid density = p 
Fluid viscosity = m 

x s = air pressure p absolute 
x r = manometer height 

Exercise 1.33 

1.34. If the general form of a transfer function is 

T(s) = 

a s m 4- a^™' 1 + 

b s n + V n_1 + ••■*« 
where a's and 6's are real, 6 ^ 0> and n > m show that 

Re [rtfo),)] = *((*>,) 

is an even function of co, and 

Im [TXfev)] = /(co,) 

is an odd function of to,. Furthermore 

lim -^- < oo and lim -^ = 

V-0 to/ 

co _> co w / 



1.35. A heavy disk with a diametric groove rotates in a vertical plane. A 
small mass particle m slides without friction in the groove under the influence of 
gravity force, centrifugal force, and the force of a linear spring in the groove of 
constant k. The disk is driven by external means to rotate at a constant speed 

(a) Show that if we call k\m = w 2 , for co f 2 < w 2 /2, the path traced by the 
particle in space is a circle lying in the lower half of the disk and for 

CO* > COS > CO 

72 the circle lies in the upper half of the disk. 

(b) What happens when co f 2 = co 2 /2 and co f 2 > co 2 ? 

Exercise 1.35 

1.36. It is known that 


2ttI J e . 

ds = e ai 

in which c is any real number greater than the real part of complex number a 

c > Re (a) 
(a) Use partial fractions to show that 

e pt — e yt 



fc + ioo 
Jc — ico (■ 



e st 


Jc — ico \S 

- P)(s - y) 


s + a 

. e st ds - 

(a + 0)e/» - (a + y)e** 

[s - Ms - y) P-V 

where c is any real number greater than the real parts of both $ and y. 


(6) Use the results in (a) to show that 

_i_ r +<o ° g ds = 1 

2tj7 J c _i oo S 2 + CO 2 ^ CO 

sin cof 

2tt/ J c-i co s' + CO^ CO 


2wi J c 

C + 100 Q * 


ds = cos co/ 

c— too s -\- oy 

where a> is real and c > 0. 

(c) Derive (74) and (72) by (94) and the foregoing results. 

(d) Derive the impulse response for (78) by (94) and the results in (a) and 
compare with results given in Art. 1.17. 


Systems with Two 
Degrees of Freedom 


2.0 Introduction 

All of the systems studied so far contain only a single variable of 
configuration, say x(t), and are therefore called single-degree-freedom 
systems. When a system requires two variables or coordinates, say x^t) 
and x 2 (t), to specify its configuration at any instant t, it is said to have two 
degrees of freedom. 1 Such a system serves as a simple model for the study 
of the general oscillation characteristics of systems with several degrees of 
freedom. The purpose of this chapter is to introduce the reader to the 
more general analysis contained in Chapter 3. 

2.1 Free Undamped Vibration — A Model and Its Equation of Motion 

Consider the system shown in Fig. 44. It consists of two masses con- 
strained to move in a horizontal line. The masses are connected to a 
stationary frame and to each other by three linear springs. 

Using the displacements of the two masses from their respective 
equilibrium positions, x 1 and x 2 , as the coordinates, we have the following 
equations of motion for the two masses: 

Wl 11 X 1 = ^n^l ^12V C 1 X 2' 

7^22^2 == ^22^2 ^12v^2 X l) 

1 A detailed discussion of degrees of freedom and its associated number of coordinates 
is to be postponed till Chapter 3. For the present an intuitive understanding of the 
terms is sufficient. 




^2^1 T ^22^2 T \^22 ' ^12/'*' 2 == ^ 



This is a set of linear homogeneous differential equations with constant 
coefficients. In the absence of first order (or velocity) terms, it is known 
that the solution consists of linear combinations of sine and cosine 




x x 


Figure 44 

x 2 



functions with four constants of integration to be determined by the four 
initial conditions of the system, which specify the displacements and the 
velocities of the two masses at t — 0. Standard methods for obtaining the 
solution are available. However, we shall take a physical approach that 
will help us to understand the problem better. 

2.2 Principal or Normal Modes 

Before proceeding with our analysis, let us recall that of the three 
quantities that characterize the free vibration of a single-degree-freedom 
system, only the frequency is an inherent property of the system, whereas 
the other two (amplitude and time-phase angle) are determined by how and 
when the system is set into motion by external agents. Consequently, it 
interests us here to find out if there is also some behavior mode that is 
inherent in a system with two degrees of freedom and is independent of the 
manner in which the system is put into motion. 

Since the system is conservative, a periodic motion may be possible. 
This leads us to ask the question, "Can the system vibrate in such a way 
that the motions of the two masses are simple harmonic motions of equal 

Assuming that this is possible, we set as usual 

x 1 = ?. 1 e' 




in which ^ and X 2 are complex amplitudes. The substitution of (2) into 
(1) results in a set of linear homogeneous equations in Xy and /. 2 . 

(-ra n co 2 + k u + k 12 )X x - k 12 ?. 2 = 

-k x2 l x + {-m 22 o 2 + k 22 -f ^12)^2 = 

It is possible to satisfy (3), hence (1), other than by choosing the trivial 
solution X x = X 2 = 0, if and only if the determinant of the coefficients 
vanishes; that is, 

■m u aj 2 -f- k u + k 12 —k 

—k 12 —m 22 oj 2 + k 22 + k 



= (4) 

This is a quadratic equation in a> 2 and is called the frequency or character- 
istic equation of the system. Its roots are determined only by the constants 
of the system. Simple algebraic operation shows further that the two roots 
of co 2 are real, nonnegative, and distinct. Hence we conclude the following; 

(i) The system is capable of motions describable by the equations in (2). 
where co is real, hence the motions of the two masses are simple harmonic 
in nature and have the same frequency. 

(ii) There are two possible values of co that satisfy the conditions 
prescribed. These become the natural frequencies 2 of the system. To each 
natural frequency there corresponds an amplitude ratio /u = Ag/A^ 

m n a) 2 + k n + k 12 k 


-LJ- ■ -l-l ■ -L<i X£ fK\ 

/C 12 ^22 W ~r "^22 • ^12 

These ratios are also real. In other words, although the /'s themselves 
are not determined by the system, their relative magnitudes are ; further- 
more, since the ratio between the two is real, the displacements of the two 
masses corresponding to a given natural frequency are in phase (or 180° 
out of phase if the ratio is negative). 

2.3 General Solution 

Therefore, we have seen that there are two basic ways in which the 
system can vibrate; they are called the principal modes of vibration. 
Each is characterized by a nature frequency co and a corresponding 
amplitude ratio ju. These two modes can operate alone; and. since the 

2 As before, we call co frequency for short, instead of the more exact term circular 
frequency, when there is no ambiguity involved. 


system is a linear system in which superposition rule holds, the two modes 
can operate simultaneously yet independently. Let the two natural 
frequencies be co' and co" and the corresponding amplitude ratios, p and 
/u". A possible solution of the equations in (1) is 

Xl (t) = X^e iM ' 1 + X{e iuj " 1 


x 2 (t) = iiix'e 1 ™ 1 + vyv"" 

in which A/ and X{ are two arbitrary complex numbers. Using standard 
representation, we have 

x x {t) = C ' cos (oj't - a') + C " cos (a/7 - a") 

x 2 (t) = fi 'C ' cos {co't — a') + ^"Cq cos {oft — a") 

in which C ', C ", a', and a" are the absolute values and the arguments 
of 2/ and A/', respectively. Altogether there are four arbitrary constants, 
or constants of integration, which may be adjusted to fit any initial con- 
ditions. Thus (7) is the general solution of the differential equations of 
motion, since it satisfies the equations and has the necessary number of 
constants of integration. To give expressions of these constants, in terms 
of the initial displacements and velocities of the two masses, involves only 
routine algebraic operations, which can be dispensed with here. 

The use of the displacements from the equilibrium positions of the 
masses as the variables of configuration is arbitrary though convenient. 
Any pair of independent geometrical quantities, which specifies the 
location of the two masses, can be used as the variables of configuration. 
We call a pair of such variables the generalized coordinates of the system. 
In particular, any two independent linear combinations of x x (t) and x 2 (t) 
can be chosen as the coordinates. 

Suppose we choose a set of coordinates p', p", to satisfy 

X l = P + P P = -J-7, 7T 

or (8) 

„ „ „ (px 1 - x 2 ) 
x 2 = pp + pp p =— — 

(/* - 11 ) 

Then, from (7), 

p'(t) — C ' cos (co't — oc') 

p"(t) = C " cos (co't — oc") 





Thus p and p" are two sinusoidal harmonic functions, each of which has 
its own amplitude, 3 frequency, and time-phase angle ; they are called the 
principal coordinates and represent the principal modes of vibration of the 

If we had prior knowledge of // and p" and chose at the very beginning 
p' and p" as the coordinates, we would have obtained two differential 
equations of motion for the system, with each equation having only one 








J *i 





+-rn'— i 

< — ■ — m" 

— >- 


-< — n' - 




Figure 45 

variable. Since their solution is known to be given by (9), they would be of 
the form (see Exercise 2-4) 

p' + (co') 2 p' = 

p" + (co")Y = 

In this way the two degrees of freedom are mathematically separated. 

Because of the lack of clairvoyance in the values of ju' and p" we cannot 
take advantage of this transformation of coordinates from k's to /?'s at the 
very beginning. However, the existence of a transformation that reduces 
(1) into (10) is of importance to our later discussions. 

A mechanical model for the separation of the two modes can also be 
conceived. In Fig. 45 we have two spring-mass systems of natural frequen- 
cies a/ and to". Two weightless levers are attached to the masses, as shown. 
For small oscillations the displacements of the masses are given by (9). 

3 It will become evident that the amplitude of p's has no intrinsic physical meaning 
because in defining p's arbitrary multiplication factors may be introduced. 


Let X x and X 2 be two points located somewhere on the levers, as shown. 
The displacements of these points are then 

YYl ryi' 

x 1 (t) = T p'( t ) + T p"(t) 
Xz(t) = n - P '(t)- n -p"(t) 

Evidently, by making 




-" = m 


these equations are equivalent to (7), except for a multiplication factor. 

In other words, with proper choices of a)', to", rijm', and n'jm", the 
free vibrations of the system in Fig. 45 can be made to simulate those of 
any undamped systems having two degrees of freedom. 

2.4 Formulation by Energy Consideration — A General Analysis for the 
Free Vibration of Systems with Two Degrees of Freedom 

The two differential equations in (1) were set up by considering the 
dynamic equilibrium of the masses. An alternate approach is by consider- 
ing the conservation of energy in the system. This approach is often more 
convenient for systems with complicated constraints because of the 
following reasons. The first is that energy expressions, being scalar and 
nonnegative, can be written down with less reference to the geometry or 
the kinematics of the system. The second is that constraint forces, which 
do no work, will never appear in the resulting equations, whereas they 
often appear in the equilibrium equations and have to be eliminated. 4 
The kinetic energy and the potential energy expressions for the system in 
Fig. 44 are 

T = iO"irV + ™22 f 2 2 ) 

V = 2V^ll a i T -^^12'^1^2 ~T ^22^2 ) 

in which 

C ll == ^11 "I" ^12 C 22 = ^22 "I" ^12 C 12 = — ^12 

The absence of the term x^x^ in the first expression is due to the particular 
choice of coordinates. In general, for a linear system whose configuration 
is specified by a pair of generalized coordinates q x and q 2 will have the 

4 See examples in Art. 2.11. 



following quadratic forms for the expressions of its kinetic and potential 
energies : 5 

T = i(«itfi 2 + 2a 12 q 1 q 2 + a 22 q 2 ) 

v = K<Mi 2 + 2c 12 q ± q 2 + c 22 q 2 ) 
This statement is amplified in the next chapter. In the meantime, we shall 
consider (11) as the definition of a linear system. Since the system is in 
free vibration, the principle of conservation of energy requires 

^-(T + V) = 

Upon differentiating (11) and rearranging the terms, we obtain 

(fln$i + c n?i + *12#2 + c i2 a 2)qi + Oi2?i + c 12 q x + a 22 q 2 + c 22 q 2 )q 2 = 

This relation must hold for all possible vibrations of the system and at all 
times. However, the values of q x and q 2 at a given instant are not related 
by dynamics to the rest of the terms. 6 For instance, the system can be set 
into motion with arbitrary combinations of q 10 , q 20 , q 1Q , and q 20 . Hence 
the coefficients of q x and q 2 in (12) must vanish independently, or 

<hfi\ + c n q x + a 12 q 2 + c 12 q 2 = 

fl 12& + C 12^1 + fl 2 2^2 + C 22^2 = ° 

This is the set of differential equations that describes the motion of the 
system. 7 These equations contain the same set of coefficients as the energy 
expressions. The a 12 terms represent the inertia coupling between the 
coordinates, and the c 12 terms represent the elastic coupling. 8 

By either of the two approaches illustrated, we find that the frequency 
equation of the system is 

— a 12 co 2 + c 12 

2 I n —n ,.£ 

= (14) 

— a 12 (o* + c 12 —a^oj* + c 22 
(a u a 22 — fl 12 2 )co 4 — (a n c 22 + c n a 22 — 2a 12 c 12 )w 2 + (c n c 22 — c 12 2 ) = 

5 The change of the notation from the aw's to the as and from the x's to the </*s is in 
keeping with the common usage. The a's and c's are called the inertia constants and the 
elastic constants, respectively. Note also the appearance of the term a 12 in the general 

6 In contrast, the displacement </'s produce spring forces that in turn produce accelera- 
tion <7"'s and are thereby dynamically related to the accelerations. 

7 Those who are familiar with Lagrange's equation of motion can easily verify that 
(11) leads to (13). 

8 Note that the presence or absence of coupling terms depends on the choice of 
coordinates. When no coupling terms appear, the coordinates become the principal 




This quadratic equation of co 2 can be solved in the routine manner. 
However, a particularly interesting representation of its solution is given 

Let us define 

\ x u 12 

a = 

co„ 2 = 








[ 12 

L ll 
C 12 

f 22 






• 2 2 




It can then be verified that the two roots of co 2 in (14) are 


= 21! 



iKl 2 + "22 2 ) ± \W 




) 2 + 4K 2 2 ) 2 ] 


This is the familiar expression for the principal stresses in plane stress 
problems. Hence Mohr's construction, shown in Fig. 46, can be used here 
as well. The reason for bringing Mohr's circle into this discussion is to 
point out that there is an intimate mathematical connection between the 
analysis of the principal modes of vibration and of the principal stresses, 
although the problems are physically unrelated. 9 Incidentally, the ex- 
pressions in (15) will not look so complicated if the inertia coupling is 
absent, or a 12 = 0. In that case we may take the system in Fig. 44 as a 
model and find 





2 _ 


Substituting (17) into (5) yields 

OT n W 22 


772 2 2^2 





Vm^ -W12 2 W22 2 - c ° 2 

From Fig. 46 it can be seen that the amplitude ratios corresponding to the 
two principal modes can be represented by the tangents of the angles d' 
and 0" indicated 

where 6' + d" = 90°. 

V "jd^ = tan 0' 

Vt7? 22 A 2 " 

= —tan 



9 Another problem having the same mathematical connection with these two is the 
one dealing with the principal moments of inertia about centroidal axes. 

10 The negative sign stems from the fact that co is less than either co n or co 2 2- 




Thus in the absence of inertia coupling the principal modes of vibration 
can be represented by two perpendicular vectors in a proper coordinate 
system, as shown in Fig. 47. A more generalized orthogonality relation- 

Figure 46 

Figure 47 

ship also exists in systems containing the "inertia coupling" terms, as 
discussed in Chapter 3. 

2.5 The Use of Influence Coefficients 

The elastic property of a system is often more conveniently expressed 
by a set of quantities called the influence coefficients. The influence coeffi- 
cient y 12 is defined as the displacement of the system at point 1 owing to a 




unit force applied at point 2. 11 The coefficients, y n , y 21 , y 22 , etc., are 
similarly defined. It is stipulated, however, that at a given point all dis- 
placements have to be measured in the same direction as the unit forces 
applied at this point, although they may not be in the same direction of the 
displacements and the forces at the other point. The meanings of the 
words "displacements" and "force" are generalized later to cover other 
things. At present, it suffices to use them in their restricted sense. 

Figure 48 

To illustrate the use of influence coefficients in vibration analysis, let us 
consider a system consisting of a weightless beam supported in some way 
and carrying two concentrated masses at points 1 and 2, as shown in 
Fig. 48. Let q 1 and q 2 be the transverse displacements of the beam at the 
location of the masses at the instant t. Taking the beam as a free body, we 
see that the two masses exert inertia forces of m n q 1 and m 22 q 2 to the 
beam. 12 These forces with the necessary reactions at the supports produce 
q x and q 2 . 


-q x = y^m sl q 1 + y^m 22 q 2 

-q 2 = y ZL m 11 q 1 + y 22 m 22 q 2 

By expressing the q's in terms of the q's, we have 


m V&\ = ~ (-722^1 + 7l2? 2 ) 




22?2 =-(-yil#2 + 721?l) 

11 It is implied here that the system is supported in such a way that forces can be 
applied to it without producing accelerations; otherwise, the influence coefficients 
cannot be immediately defined. (See Art. 2.9.) 

12 With the beam as the free body under consideration, the forces exerted by the 
masses are opposite to the accelerations of the masses. The forces shown in Fig. 48 
are in their positive directions consistent with the positive directions of the displacements. 
The actual directions of the forces are in the reverse, but this fact is to come from the 
solution of the problem and need not be assumed beforehand. This is very elementary, 
although at times confusing. 






y = 

7n 7i2 

721 722 

By comparing (21) with (1) and recalling the definition of the spring 
constants c's, we see that 

Ci, = 

Co = 





C 12 — 






Incidentally, we have also verified Maxwell's well-known reciprocal 
theorem which states 

7l2 = 721 

The relationship between the influence coefficients and the elastic 
constants of a system, as represented by (22), has certain symmetry. We 
can easily verify that 

c 22 

m = - 

in which 


c = 

7i2 = - 







Thus in the language of matrix algebra the influence coefficients and the 
elastic constants of a system form matrices that are inverse to each other. 

The transformation of (20) into (21) shows that it is not often easy to 
say what kinds of coupling exist between the coordinates by looking at 
the differential equation. In this problem there is no inertia coupling 
between q 1 and q 2 , even though both accelerations appear in both equations 
of (20). This is obvious if we write down the kinetic energy expression for 
the system and observe the absence of a q x q z term. 

Equation (20) can be directly solved without converting the influence 
coefficients into the elastic constants. Setting, as usual, 

in (20), we have, after multiplying the equations by — 1/co 2 , 


^ll7ll - — 2 J^l + ("^2712^2 = 
(^Il7l2)^l + (^22722 - ^P 2 = 



The frequency equation is then obtained by the condition for the existence 
of nontrivial solutions, namely, the vanishing of the determinant 


m ii7n 2 m 22/l2 

"Wl2 ^22722 2 

CO 1 

= (25) 

2.6 Rayleigh's Quotient 

In dealing with systems having a single degree of freedom, it was found 
that the natural frequencies can be obtained directly by equating the 
maximum values of the two energies. 

T = V 

We shall now determine where this procedure will lead us in dealing with 
systems having two degrees of freedom. Let us assume that the system 
vibrates in one of its principal modes with 

q 1 = X x cos (cot — a) q 2 = X 2 cos (cot — a) 

Hence 13 

T m = i^ViA 2 + 2a 12 V 2 + a 2 2^2 2 ) =/^ 2 
V m = ifoA 2 + 2c 12 A 1 A 2 + c 22 k 2 2 ) 
2 V m (X x , X 2 ) 



Since V m and / are homogeneous quadratic forms of X x and A 2 , co 2 is a 
function of the amplitude ratio ju = X 2 \X V Without knowing the correct 
value of /u, co cannot be found by Rayleigh's method, and if ju is to be 
found by the method previously discussed we shall have found co already. 
The matter, however, bears further discussion. 

Let us consider the ratio of the quadratic forms as a function of ju and 
call it Rayleigh's quotient. 

n , v Cn + 2ci2^ + c 22/ w 2 

60) = r-z : 2 ( 28 ) 

Q(ju) becomes co 2 when /u is the amplitude ratio of a natural mode; other- 
wise it is always positive and finite, since the energy expressions themselves 
are always positive and finite for any finite values of A x and X 2 . It is there- 
fore bounded both from above and from below. If these bounds can be 
established, they will at least give us an estimate of the natural frequencies 
of the system. 

13 Aj and A 2 are real quantities in this discussion. 



It is not difficult to reason out that a plot of g-versus-// relationship 
will result in a curve having the features shown in Fig. 49. Rayleigh's 
quotient thus has two stationary values, one a maximum and the other a 
minimum. These can be located by setting 

which is equivalent to 

dQ dO 

9IT ar 2 = (29) 

Substituting (27) for £) m (29), we have 

bq u,dv m wr dp 

£M = 1/£^_ F _2_ = o 

3A, p\ J dl, m dXj 

dQ \ PdV m rr df 

Since Qf= T„ 

^ = _L ( f °J^-v °L =0 

Multiplication by Qf* yields 


^r-(v m -TJ = 

^(V m -T m ) = (30) 

Note that these are not trivial identities, even though T m — V m = 0. 
In combining (30) with (26), we have 

(-flu 2 2 + c u )h + (~a 12 Q 2 + c 12 )A 2 = 

(-0120 2 + *iaWi + (-^ 22 e 2 + c 22 )h = 

The condition for this set of homogeneous equations to be consistent is 
that the determinant formed by the coefficients must vanish. This con- 
dition is the frequency equation (14) exactly. 

Thus we have shown that the two stationary values of Rayleigh's quotient 
are the natural frequencies of the system. 




Perhaps it is worthwhile here to add a physical model to our analysis. 
Take again the system consisting of two masses and three springs. Let us 
incorporate into the system some device, such as a gear train, that con- 
strains the motions of the two masses to a given displacement ratio. In 
so doing we reduce the system to one with a single degree of freedom. 
Equations (26), (27), and (28), however, are still valid, except that fx = 
IJX 1 is now a given ratio and the frequency of the constrained vibration 


^ jU = 

_ A2_ 

Figure 49 

is determined by it. In general, if the ratio imposed by the constraining 
device changes a little, the frequency of the constrained vibration changes 
accordingly, But, when the ratio imposed is such that the frequency is a 
maximum or a minimum, c/Q/d/Lt = 0, a small change in the ratio produces 
a change in the frequency which is of a smaller order. The various elastic 
and inertia forces in the existing vibration are therefore in an equilibrium 
by themselves without the help of the forces exerted by the constraining 
device. Hence, if the constraints are removed, the vibration remains 
unchanged, and it is one of the normal modes of the system. 

We have shown that the natural frequencies of vibration can be found 
by maximizing or minimizing Rayleigh's quotient. However, to carry out 
the process in its entirety requires almost the same algebraic steps as those 
involved in the standard procedure. The practical value of the analysis 
just presented is then lost. To apply Rayleigh's method in a practical way, 
the amplitude ratio corresponding to a given mode instead of being 
determined exactly is only estimated. The frequency is computed from 
Rayleigh's quotient by using the estimated amplitude ratio. This procedure 
gives surprisingly good results because, as we have seen, near a natural 
frequency of the system the value of Rayleigh's quotient is insensitive to 
small changes in the amplitude ratio. (See Art. 2.12.) In systems with 




two degrees of freedom the exact value of the natural frequencies can be 
obtained without much computation. The practical utility of Rayleigh's 
method and its modification is in systems with many degrees of freedom, 
but it is more clearly illustrated when applied to a simpler system. 

2.7 Vibration of Damped Systems 

If a viscous damper of damping constant c is added between the two 
masses in Fig. 44, the term c{x 1 — x 2 ) will have to be added to the first 
equation of (1) and subtracted from the second equation of the set. 
In general, if there are viscous damping forces in the system, they will be 
proportional to q's or linear combinations of q's in a manner analogous 
to elastic forces that are proportional to the </'s or linear combinations of 
the </'s. By adding damping terms proportional to the velocities of the 
g's, the equation of motion (13) becomes 

fl ll?l + b ll<il + C ll?l + «12^2 + ^12?2 + C 12 q 2 = ° 
*12?1 + *irfl + C 12?l + «22?2 + ^22^2 + C 22 q 2 = 

The method of solution is analogous to that of solving (13); that is, 
by letting 

q 1 = X x e st or q 1 = l x e iat 


q 2 = X 2 e si or q 2 = X 2 e iat 

in (32). This leads to the frequency or characteristic equation 
a n s 2 + b n s + c n a 12 s 2 + b 12 s + c 12 
a 12 s 2 + b 12 s + c 12 a 22 s 2 + b 22 s + c 22 

which is a biquadratic equation in s or ia, hence possesses four roots: 
s l9 s 2 , s 3 , and s 4 ; the solution of (32) is then 

= 0- 

(32) 14 



q 2 = X 21 e s ^ + A 22 e**' + hz e * 

+ h^ 


in which the ratios A u /A 21 , A 12 /A 22 , ^ 13 /A 23 , and A 14 /A 24 are determined by 
s l9 s 2 , s 3 , and s 4 , respectively. 

012*1 2 + *>12*1 + C 12 

[ 22 

Si + u 22 Si + c 22 


*ii*i 2 + Vi + c n 


^12^2 ~i~ ^12^2 "1" C 12 


a n s 2 2 + & u j 2 + c u 


^12^1 2 + ^12*1 + C ia 
^22^2 ~i~ #22^2 i ^22 

a 12 s 2 2 + Z? 12 5 2 + c 12 


14 We assume here that b 12 = b 21 , a fact which is discussed further in Art. 3. \ 


The initial displacements and velocities of the system and the relations 
in (36) determine the eight constants of integration in (35). If all the roots 
of (34) are real, the ratios in (36) are real; and with real initial values of 
displacements and velocities, all the A's are real. The resulting motion is 
nonoscillatory or aperiodic, and the system is overdamped. Furthermore, 
since the damping forces dissipate energy, the motions must decay with 
time and the real roots of (34) must be negative. 

If (34) has complex roots, they must come in complex conjugate pairs, 
since all the coefficients in this algebraic equation are real. Let 

s ± = s and s 2 = s 

be a pair of such roots. Then 

e S x t _ e 8t e §4 _ gSt 

will also be complex conjugate to each other, and, in order to yield real 
values of q 1 and q 2 , 15 the first two coefficients in each of the equations in 
(35) must also be complex conjugates. 

Aq = A. A^ 2 ^ A 

If in (36) we let 

A 21 == '"^ll =: '" ^"22 == 21 == ^" 

The part of the solution represented by the first two terms of (35) is then 

qi = Xe st + le u 

q 2 = rle Ht + ~rke gt 

s = ia = io) c — y\ X = \Ce~ x% and r — /ue~ id 

s = —ko c — 7] 1 = \Ce ly - r = jue l() 

Substituting these into the expressions for q 1 and q 2 , we have 

q x = Ce~ r,t cos (co c t — a) 

q 2 = juCe~ 7lt cos (w c t — a — 6) 

If (34) has two pairs of complex roots, the solution (35) may be expressed 

q x = Ce-' 1 ' 1 cos (co c 't - a') + Ce'^ 1 cos (co c "t - a") 

q 2 = fi'Ce- 71 ' 1 cos (co c 't - a' - 6') (37) 

+ ffC'e-f' cos (co e "t - a" - 6") 

15 Note that (35) is a conventional representation for solutions of (32). Under any 
real initial conditions it must yield real solutions. 


in which — r,' ± ico c ' and —rf ± ioof are the roots of (34); p\ fi", V, 
and 0" are determined by (36); and C, C", a', and a" are determined by 
the initial conditions. 

It is evident from (37) that the oscillation of a slightly damped system 
with two degrees of freedom may be decomposed into two characteristic 
damped oscillations in a manner analogous to the decomposition of 
undamped vibrations into principal modes. In other words, by choosing a 
pair of coordinates p and p", which are certain linear combinations of 
q± and q 2 , the set of equations (32) can be transformed into another set 
in which the two variables are separated. The coefficients for this trans- 
formation are determined by (36) and will be complex numbers if the 
roots of (34) are complex. The numerical computation involved in this 
transformation is, however, very tedious. 

To sum up, the nature of free vibration of a two-degree-freedom system 
depends upon the roots of the characteristic equation (34). Since all the 
coefficients of this equation are real, it may be factored into two quadratic 
factors : 

(s 2 + Irj's + to' 2 ){s 2 + 2rfs + o/ 2 ) = 

The constant terms in the factors are the squares of the natural frequencies 
of the undamped system because, if all the &'s in (34) are zero, r( = rf = 0. 
and the equation is identical to (14). The two >/'s are the same as those in 
(37). If there are damping forces, rf and rf in general will be positive, 
although in special cases one of the two may be zero. To each of the 
quadratic factors there corresponds a mode of vibration. In each factor 
the condition ?/ < co indicates a damped oscillation with damped frequency 
co 2 = (o 2 — ?y 2 , the condition >/ = indicates a sinusoidal oscillation, 
and the condition r\ > to indicates an aperiodic motion. The condition 
co = is discussed in Art. 2.9. 

To factor the biquadratic equation, a numerical method must be used. 
For such well-known methods as Graejfe's root-squaring method and LirCs 
iteration method and its modifications the readers are referred to textbooks 
on numerical analysis. It may be added that the general method of 
solution for biquadratic equations is not convenient to use. 

2.8 Forced Vibration 

If a two-degree-freedom system is subjected to external excitation, the 
differential equations of motion will no longer be homogeneous. The 
external excitations may be forces applied to the mass particle in the 
system or in motions imposed on certain parts of the system. In either 




case, the excitations are known functions of time, and the general form for 
the differential equations of motion is 

Ml + V/l + C ll<7l + fl 12?2 + ^12^2 + C 12 q 2 = QlO) 

<*v&\ + &i2<7i + c i2<7i + a 22 q 2 + b 22 q 2 + c 22 q 2 = Q 2 (t) 


This set of equations can be solved by a number of standard methods. 
Here we will treat special cases of the most practical interest. 


It is known that periodic functions can be expanded into a Fourier 
series of sinusoidal functions. Since the system is linear and the principle 
of superposition holds, the solution for periodic forces can be built up 
from solutions for sinusoidal forces. We need therefore deal here only 
with sinusoidal forces. Let 

Qi(t) = V 

and Q 2 {t) = F 2 e lu) f l 


in which F 1 and F 2 are in general complex numbers. The steady-state 
solution must be of the form 

q 1 = ^e' 10 ' 1 and q 2 = l 2 e ni>lt 
By substituting (39) and (40) in (38), we have 

(— co f 2 a n + io) f b n + Cn)^ + (— co f 2 a 12 + uo f b 12 + c 12 )X 2 = F 1 
( — co 2 a 12 + i(o f b 12 + c 12 ))^ + { — (» 2 a 22 + Uo f b 22 + c 22 )X 2 = F 2 



This set of equations possesses a pair of complex solutions for ?. 1 and A 2 , 
which are to be interpreted in accordance with the discussion in Art. 1.7. 
They are given by 

F lt —oj f 2 a 12 + ia)fb 12 + ,c 12 

F 2 , —co f 2 a 22 + ico f b 22 + c 22 

-(o 2 a u + ko f b n + c 11 ,F 1 
—oj f 2 a 12 + icOfbji + c 12 , F 2 











D = 

co f 2 a n + ico fb n + c n , —co f 2 a 12 + ib 12 (o f + c 12 
co f 2 a 12 + i(o f b 12 + c 12 , —«) f 2 a 22 + z^co/ + ^22 


= (co/ — f^'coy — co' 2 )(o) 2 — irfcoj — co" 2 ) 

It is to be noted that if both rf and if are not equal to zero the determinant 
D cannot vanish and X 1 and X 2 are always finite. On the other hand, if 


one or both of the rfs are zero and the determinant D vanishes for one or 
two values of oo f , namely, to' and/or to", the correspondent / x and / 2 will 
in general become infinitely large. 

A frequency response curve for each coordinate in a two-degree-freedom 
system can be obtained from (42) for any appropriate set of parameters. 
It is not difficult to see that if the damping coefficients are not too large 
all the curves will have two peaks near the natural frequencies of the 
system. (See Fig. 58). 

(b) analysis of an undamped system by principal coordinates 

In Art. 2.3 (and Exercise 2.4) we have sketched a procedure whereby 
the set of homogeneous differential equations (1) may be transformed, 
through a linear transformation of coordinates, into another set of the 
form (10). Such transformation is also possible for equations in the general 
form (13) and for nonhomogeneous equations 

%?i + Cntfi + «i2^2 + c 12 q 2 = Qi(0 


<h!&\ + C 12<7l + «22^2 + f 22<72 = 2 (O 

Without the use of matrix algebra, the steps involved become very tedious. 
Since the subject is discussed more fully in Chapter 3, we give only the 
final results here. If, as in (8), we let 

ft = p' + p" 

q 2 = p'p' + p" p " 

the set of differential equations (44) can be transformed into 

p> + osy = no 
p" + yy = F\t) 


(p!'a 22 + «i 2 )2i(0 - (p"a 12 + a n )Q 2 (t) 

HO = 
HO = 

(a n a 22 - a 12 2 )(p"-fx') 

-{pa 22 + fli 2 )8i(0 + (/^ /fl i2 + a u )Q 2 (t) 
(a n a 22 - a 12 2 ){fi" - p,') 




Since the variables are separated in (46), the solutions for p and p" can be 
obtained by methods discussed in Chapter 1. The initial conditions of/)' 
and p" are determined by those for q 1 and q 2 through (45). Similarly, the 
solutions for q x and q 2 are obtained by substituting the solutions for p 
and/?" into (45). 


Of particular interest are cases in which there is no inertia coupling 
between the coordinates. For such systems a n = m n , a 12 = 0, a 22 = m 22 , 
and, according to (19), 

' " . " hl n 
ju /u H = 


The expressions of F' and F" can then be simplified to read 

F'(t) = —JL Ql ( t ) + p'QJj) 

m n(f* - P ) 


no = —t4 — k aw + /wo 

m n {/4, — ix ) 
m n (/Li" — fif)(p' + co'Y) = fi'Q^t) + /x'ju' f Q 2 (t) 



According to the principle of superposition discussed in Art. 1.11, 
the solutions to (46) or (49) can be obtained by substituting the appro- 
priate parameters of these equations into (66), (74), and (76) of Chapter 1. 
This method is particularly advantageous when transient solutions are 


When the system is damped, separation of variables by linear transform 
is possible but not practical. Nonperiodic motion of a damped system 
produced by external forces is of great interest in the study of control 
systems. The problem is best treated by methods in operational calculus, 
which is a subject by itself and cannot be included here. On the other 
hand, the same problem is only of passing interest in theory of mechanical 
vibration because complex systems with many damping forces are rare. 
The simpler practical problems can usually be solved by more elementary 
methods without too much extra labor. 

If a transient problem is to be solved, the method discussed in Art. 
1.11(d) may be used. First we obtain the transfer function in steady-state 
analysis. For instance, according to (42), the transfer function from Q x 
to q x or the frequency response of q 1 to Q x in (38) is 

^ , x a ii s + b 22 s + c 22 

Tjs) - — m — m 

where D(s) may be factored into 

D(s) = a(s - s x )(s - s 2 )(s - s 3 )(s - s 4 ) (51) 


in which a = a n a 22 — a 12 2 , and s t , s 2 , s z , and s 4 are the roots of (34). 
If none of these roots is pure imaginary, the impulse response is then 

^4f'S!^fH|^ (52 ) 

ZTTlJ-icc D(S) 

and the response <7 n (0, due to an arbitrary Q x (t) applied at t = when 
the system is at rest and in equilibrium, becomes 

fci(0 = 7~. " 2 " e^'^ ds) Q x {t) dr 

LTTl JO \J-»oo D{S) I 

Should D(s) vanish for some pure imaginary values of 5. a real positive 
constant c must be added to the limits of the integral with respect to s. 
The impulse response function can be evaluated by the results in Exercise 
1.35 after the numerator and denominator of the transfer function T(s) 
are factored and the fraction is broken into partial fractions. 

2.9 Degenerated Cases 

(a) the case of zero natural frequency- 
semidefinite system 

In most vibratory systems the vibrating masses are connected with a 
stationary frame by elastic and constraining forces in such a way that the 
equilibrium configuration of the system is uniquely determined and the 
system can remain at this configuration only when there is no motion. 
There is, however, a class of systems to which this does not apply. The 
inertia elements in these systems may move as a rigid-body ensemble 
without disturbing the equilibrium of the forces acting upon them. Such 
systems are said to be semide finite. A simple example of a semidefinite 
system is the one obtained by removing the two outer springs k n and k 22 
in Fig. 44. This system may be considered as a model of such practical 
things as two railroad cars elastically coupled or two pulleys mounted on a 
rotating shaft. 

To analyze the vibration of a semidefinite system, two approaches are 
possible. We can treat it like any other vibratory system by writing down 
the differential equations of motion and obtaining the frequency equation. 
In the process of doing so we will find two peculiarities; the first is that the 
influence coefficients of a semidefinite system are not all defined, and the 
second is that the frequency equation will have a zero root, or roots, which 
corresponds to the possible rigid-body motions. The difficulties introduced 
by these peculiarities are discussed more fully in Chapter 3. The other way 
of analyzing a semidefinite system is by eliminating the rigid-body motions 




with the assumption that the total momentum vectors (linear or angular) 
associated with these rigid-body motions are identically zero. This is equi- 
valent to introducing constraints into the system and thereby reducing the 
number of degrees of freedom. For instance, in Fig. 50 a rotating shaft 
carrying three disks, representing a driving pulley, a driven pulley, and a 

Figure 50 

flywheel, has three degrees of freedom. One of the three is a rigid-body 
rotation, which is of no interest to vibration study. The set of differential 
equations is originally 

/A + M&1 - 2 ) = 

/A + k 12 (d 2 - 0j) + k 23 (d 2 - 6> 3 ) = o 

I A + M^S - 2 ) = 

Adding the three equations together, we have 

/A + hh + 1 A = o 



/A + h®2, + ^A = constant 

By assuming that there is no rigid-body motion, this constant becomes 
zero, or 

IA + I 2 2 + / 


constant = 

This last constant can also be set equal to zero as the 0's are taken to 
be zero when the shaft is untwisted. By substituting 

A#l + 7 3^3 
®2= J 


into the first and third equations of (53), we have 

i A + *„(«, + /A + /A ) = o 

This set of equations can be reduced to the standard form of (13) by 
multiplying them with the factors 

h /^23 „ i h /^12 

7 3 ^12 *1 ^23 


This procedure of eliminating a number of variables and equations 
according to the number of possible rigid-body motions is, however, not 
always convenient in a more complex system. 

(b) the case of zero mass — "half" degree of freedom 

We have seen that the motion of a single-degree-freedom system is 
governed by a differential equation of the second order and that of a two- 
degree-freedom system is governed by two differential equations of the 
second order, which are equivalent to one differential equation of the 
fourth order. If, however, the inertia forces in one of the two degrees of 
freedom are zero or negligible, the resulting differential equation of motion 
becomes a third-order one. Consider the system shown in Fig. 51, which 
has a single rigid body constrained to translate along a straight line. In a 
sense this system has only one degree of freedom, since the displacement 
function X(t) specifies the location of all mass particles in the system. 
However, when the system is in motion, the value of X(i) does not specify 
the location of the point P, whose motion must be considered as another 
independent unknown. This system is in reality the degenerated case of a 
two-degree-freedom system, which is obtained by first putting a mass m 
at P and then letting this mass become zero. The differential equations of 
motion degenerate into 

MX + KX + k(X - x) = 

ex + k(x - X) = 
By differentiating the first equation to obtain the additional equation 

MX + KX + kX - kx = 
and by eliminating x and x from the three equations, we have 

°^X + MX + (l + j)c* + KX= 


This is a third-order differential equation whose characteristic equation is 

— s* + Ms 2 +1 1 +-t)« + a: = 

This is a special case of the more general equation obtained by letting 
a 12 = a 22 = in (34). Although the term makes very little physical 




x/\/\/\/\/ N — ¥ — V\/\/\ 


Figure 51 

sense, we may conveniently think of such a system as having one and one 
half degrees of freedom. It is interesting to note that since a cubic equation 
with real positive coefficients has at least one real negative root at least one 
mode of the motion is an exponential decay. 

X(t) = Ae~ st 

2.10 Repeated Roots in Frequency Equations — 
Transverse Vibration of Rotating Shafts 

When the frequency equation (14) has two equal roots, the two modes of 
vibration have the same frequency, and Mohr's circle, shown in Fig. 46, 
degenerates into a point. The physical implication of this condition is 
that any set of coordinates chosen to describe the two degrees of freedom 
can be considered as principal coordinates in the sense that there are no 
couplings between them. Consequently, such orthogonality relationship, 
as described by (19), may or may not exist. 

The analytical peculiarities introduced by the existence of repeated roots 
in frequency equations are discussed in Chapter 3. At present, we shall 
study a practical problem related to this subject. 

Take the simple case of a circular shaft carrying a heavy disk in the 
center and supported by bearings at the ends. Disregarding the rotational 
freedom of the shaft for the moment, we note that the system has two 
degrees of freedom which can be described by the displacement vector of 
the disk center from its equilibrium position. Evidently the disk can be set 
into a simple harmonic motion along any direction perpendicular to the 
axis of the shaft. Hence any such vibration is a principal mode of vibration. 


For convenience, let us take the displacement components along two 
mutually perpendicular directions as the coordinates of the system. The 
differential equations for free vibrations are 

mx + kx = 
my + ky = 


in which m is the mass of the disk and k is the restoring force per unit 
deflection of the shaft at the location of the disk. The solution to this 
set of equations is 

x = x cos (cot — a) 

y = y cos (ojt - p) 

It is not difficult to show by analytical geometry that if ojt is eliminated 

between the two equations the resulting relationship between x and y 

represents an ellipse. In other words, the path traced by the disk center 

is, in general, an ellipse. For the particular case in which x = y and 

a — ft = ±7r/2, the path becomes a circle. The motion of the disk as a 

whole is a translation. That is to say, the paths traced by all points of the 

disk are congruent to one another, whether they are ellipses or circles. 

For the sake of clarity in later discussions, let us coin the term whirling 16 to 

denote the translation of the disk along a circular path. This whirling 

motion may be excited by a force which is constant in magnitude but 

which rotates with a uniform angular velocity co f . Let the force be in the 

x direction at t = and have a constant magnitude F . Its x- and y- 

components at any time t are then F cos co f t and F sin co f t, respectively. 

When this force is applied at the center of the disk, the differential equations 

of motion are 

mx + kx = F n cos to J 


my + ky = F sin co f t 

According to Art. 1.6, for co f ^ co, the steady-state solution of (56) is 


*[i-K> 2 )] 


y = 

Fo 1 

k [i - k> 2 )] 

This solution represents a whirling motion. Let us observe that in this 
motion the displacement of the disk center is always colinear with the 

16 There is no standardized terminology for this motion. Some authors use the term 
revolving or revolution for this purpose. 




force and that the two vectors are in the same or the opposite direction 
according to whether co f is less or greater than co. 

A rotating force such as the one described is realized physically when 
the disk has an eccentricity and rotates with the shaft at the speed oj f . 
Intuitively, it is reasonable to suggest that the rotation and the eccentricity 
together produce a centrifugal force that is entirely equivalent to an 



Figure 52 

externally applied rotation force. To analyze the situation properly, 
let us refer to Fig. 52. In this figure O is the position occupied by the 
center of the shaft when the shaft is undeformed, C is the displaced 
location of the shaft center, and M is the location of the center of mass of 
the disk. The vector OC represents the translation of the disk and the 
rotation of the vector CM, which has a fixed length equal to the eccentricity 
e, represents the rotation of the disk. The acceleration of the mass center 
M can be resolved into the relative acceleration of M with respect to C 
and the acceleration of C with respect to O, which is fixed. The corre- 
sponding inertia forces are shown in Fig. 52a. In addition to these forces, 
there is an elastic restoring force exerted by the shaft on the disk at C. 
Assuming that at t — 0, CM is parallel to the x-direction, we have 

mx + kx = mco f 2 e cos w f t 
my + ky = mco f 2 e sin co f t 


A comparison with (55) and (56) reveals that the displacement of the disk 
center C is given by 

x — r cos io f t 

y = r sin ojA 



mco f 2 e oj f 2 e 

k[\ - k>2)] 


and co f ^ co. 

The disk therefore whirls and rotates at the same time. The shaft 
center C whirls around O with the same angular velocity as the rotation 
of the disk around C. The over-all motion is a rotation of the disk about 
the fixed point O. In other words the angle OCM is fixed. According to 
our previous discussion, this angle is either or 180°, 17 depending upon 
whether co f is greater or less than co. 

From a practical point of view, for any co f that is different from a>, the 
displacement vector OC can be made arbitrarily small by reducing the 
eccentricity e. In other words, the whirling motion can be almost entirely 
eliminated if the disk is very well balanced. However, as co f approaches to, 
a very small eccentricity can still produce a large whirling motion if co f 
is sufficiently close to co. This speed co is therefore called the critical 
speed. The phenomenon is merely a special case of forced vibration at 
resonance, and the critical speed corresponds to the natural frequency 
of the system. 

In engineering applications the rotating speed of a shaft is always kept 
apart from its critical speed to avoid excessive vibration. In many high- 
speed devices the operating speed is above the critical speed. To reach 
that operating speed, it is necessary during starting to pass through the 
critical speed quickly. The behavior of the system near its resonance 
condition is therefore also of practical interest. In Art. 1.6 it was pointed 
out that if there is no damping steady-state oscillation is not possible at 
o) f = co. Instead, the amplitude tends to build up as shown in Fig. 12. 
In an actual case there is always some damping, and the amplitude 
eventually becomes constant at some large but finite value. We also know 
that regardless of whether or not a steady-state condition is reached the 
excitation force leads the displacement by 90° at co f = co, so that the 
relative location of O, C, and M is as shown in Fig. 52. Thus the centrif- 
ugal force of whirling motion, mco 2 r, and that of rotation, mco 2 e, are 
perpendicular to each other. In addition, there is the elastic restoring 
force kr along OC and another force F c along CM. This force F c may be 
interpreted in two different ways. 

First, let us assume that whirling has just started and that the damping 
force is negligible, so that the amplitude, or r in this case, begins to grow. 

17 If this fact is taken for granted beforehand, (58) can be derived by simple equi- 
librium consideration, as it is done in most textbooks. 


This means that the angular momentum about increases; hence there is 
an inertia force due to Coriolis acceleration. 

F r r = — [mcoir 2 + e 2 )] = Imcor — 
dt l dt 

According to discussions in Arts. 1 .6 and 1 .9, drjdt = constant. Utilizing 
(35) of Chapter 1, we have 

dr d st mco 2 e coe 

Jt = ~2 CO= ~2F° J= ^ 

F c = mco 2 e = ke 

Or we can assume that the steady-state condition is reached by having an 
amplitude large enough to produce a damping force 

F c as ccor = mco 2 e = ke 

Between these two extreme cases both types of forces exist, but they will 
not be at their maximum values. Since the forces shown in Fig. 526 are 
the only forces on the disk. Newton's second law demands that 

F c = ke = ccor + Imcofr 

These forces, however, do form a pure moment that must be balanced by 
an external driving torque on the disk in order to maintain the condition 
o) f = co. This torque is 

M t = F c r = ker 

Now if we want to speed up the shaft quickly, we must have a driving 
torque that is substantially greater than M t . This is, of course, not a 
quantitatively useful statement, since e is usually unknown and r varies 
with / before the steady-state condition is achieved, But it does point out 
the importance of having sufficient driving torque to speed up the rotation 
quickly before large amplitudes can be built up to make the torque needed 
for acceleration still larger. 

In the foregoing analysis we assume that the only inertia force is that due 
to a concentrated mass. This assumption is made merely to simplify the 
illustration of the principle that the critical speeds of rotating shafts are 
the same as their natural frequencies in transverse vibrations. If a shaft 
carries several masses, it will have several critical speeds, which are to be 
found by the method discussed in Chapter 3. If the mass of the shaft 


itself is not negligible, its critical speeds are to be determined by methods 
discussed in Chapter 4. If the location of the disk on the shaft is such that 
during vibration the disk also rotates about a diameter perpendicular to 
the plane of the shaft deflection, then the rotatory inertia of the disk 
about its diameter must also be taken into account. This effect is discussed 
in Art. 2.14. 

In short, if the entire system— the disks, the shaft, and the bearing 
supports — has an axial symmetry, a rotating force can excite two trans- 
verse vibrations that are 90° out of phase and along two perpendicular 
planes. The rotation of the shaft merely furnishes the excitation needed 
without altering the vibrational characteristics of the system in other ways. 
In other words, if the shaft is prevented from rotating while the disks are 
allowed to rotate on the shaft, the result is assumed to be the same. 
There are, however, circumstances in which the rotation produces excita- 
tion other than a centrifugal force that rotates with the speed of the shaft. 
These are discussed in Art. 2.14. 

2.11 Illustrative Examples 

In most cases the differential equations of motion for a vibratory 
system can be written down without much preliminary analysis. But when 
there are forces of constraint, which must be eliminated, the preliminary 
analysis may become somewhat involved. In this article we illustrate the 
procedure by analyzing two examples. The systems chosen have no 
practical utility, but they serve our purpose well. 

Consider first the system shown in Fig. 53. It consists of a wedge of 
mass M sliding on a smooth table and a small block of mass m sliding 
on the wedge. Two linear springs of constants k and K couple the two 
masses together and to a stationary frame. Evidently this system has two 
degrees of freedom, and a convenient choice of coordinates consists of the 
horizontal displacement of M and the relative displacement of m on M. 
Let us call these displacements x x and x 2 , respectively, and assign them 
the value zero when the gravity forces are in equilibrium with spring 
forces and forces between contacting surfaces. In this problem, the effect 
of gravity does not change with x 1 and x 2 , and we need to consider only 
the variations in these forces from their values at equilibrium configura- 
tion when we analyze the motion of the system. With this simplification, 
the free-body diagrams of the two rigid bodies of the system are as shown. 
The assumption is also made that these variations are small, so that the 


string will remain taut and the block will remain on the incline. The 
pertinent equilibrium equations are 

TiF x = K{x x — x 2 ) — kx 2 = mx 2 — mx\ cos 

SFtf = mx\ sin d = N 

2F = K(x x - x 2 ) + Mx\ - [K(x 1 - x 2 ) - kx 2 ] cos d + A^sin (9 = 


Figure 53 

By substituting the first two equations into the third, we obtain 

K(x ± — x 2 ) + Mx\ — mx 2 cos 6 + mx\ — 

A rearrangement of the first and the last of the foregoing equations yields 

(M + rri)x\ + Kx x — m cos Qx 2 — Kx 2 = 
— m cos 6x\ — Kx 1 + mx 2 + (K + k)x 2 = 


This rearrangement reduces the equations to the standard form 18 of (13). 
There are both inertia coupling and elastic coupling between the co- 
ordinates represented by the coefficients (— racosfl) and (— K), respec- 

Next we want to show how the same set of equations can be obtained 
with much less labor and thought. It can easily be seen that the kinetic 
energy and potential energy expressions of the system are 

T = \Mx 2 + M(*i ~ * 2 cos df + x 2 2 sin 2 6] 
V = \K{x Y - x 2 f + \kx* 

By simplifying and rearranging the terms in these expressions, we have 

T — \[(M + m)x^ — 2m cos 6x 1 x 1 + mx 2 2 ] 
V = \[Kx 2 - 2Kx 1 x 2 + (K+ k)x 2 ] 

A comparison with (11) reveals that 

a n = M + m a 12 = — m cos 6 a 22 = m 

c n — K c 12 = — K c 22 = K + k 

By putting these constants into (13) and (14), we can write down the 
equilibrium equation and the frequency equations immediately. 

For our second example let us consider the system shown in Fig. 54. 
A disk of mass M, moment of inertia /, and radius R rolls without slipping 
inside a cylindrical surface of radius 3R. The disk is in vertical plane and 
has a diametric slot in which a spring-mounted mass m can slide without 
friction. This slot is at a horizontal position when the disk is at its lowest 
point. The springs connected with m have an over-all constant k, and the 
neutral position of the mass is at the center. 

This system contains many constraining forces in which we are not 
primarily interested. Analysis by energy consideration will relieve us of 
the necessity of dealing with these forces. Let us choose as the coordinates 
for this system the angle 6 and the distance r, as indicated. 

18 When the equations are obtained by equilibrium consideration, they are not always 
in the standard form, but they can always be reduced to such form by simple algebraic 
operations. It is most convenient to have the equations in the standard form because 
their solutions can then be written down by direct substitution of constants into the 
standard solution. 


In terms of these coordinates, we have 

v = velocity of the center of the disk = 2R0 
Q. = angular velocity of the disk = 26 
v m 2 = velocity of m squared 

= (v cos 3d + ff + sin 3(9 - rQ) 2 

= v 2 + f 2 + r 2 Q? + 2i; r cos 3(9 - 2^ r Q sin 3(9 


Figure 54 

For small oscillations both r/R and 6 are small, so that the third and the 
last terms can be neglected. After eliminating v , we have 

v m 2 = 4R 2 6 2 + r 2 + 4Rf6 

The kinetic energy of the system is then 

T = iMv<? + ¥& + im^ 2 

= i[4(M,R 2 + mR 2 + /)0 2 + 4m/?r0 + mf 2 } 

Now let 

?/ = the rise of the center of the disk = 2R{\ — cos 0) 

= Rd 2 
y m = the rise of m = y — r sin 2d 

= R6 2 - 2rd 


The potential energy is then 

V = Mgy + mgy m + \kr 2 

= i[(M + m)Rgd 2 - 4mgrd + kr 2 ] 

From these energy expressions it follows that the differential equations of 
motion are 

4(MR 2 + mr 2 + 1)6 + 2(Af + m)#0 + 2m£r - 2m^r = 

2mRd — 2mgd + mr + kr = 

2.12 Application of Rayleigh's Method 

(a) variation of rayleigh's quotient with amplitude ratio 

Rayleigh's method is seldom used to analyze a two-degree-freedom 
system, for which solution by the exact method is rather simple. To apply 
this method to a simpler system does, however, give us the necessary 
feeling of the method. First let us describe a method that is not really 
Rayleigh's method. Take the system in Fig. 44 and assign the values 

m n = 1 m 22 = 3 
k u = 2 k 12 = 3 and k 22 = 1 
all in their appropriate units. The expression for Rayleigh's quotient is 

(k n + k 12 )X x 2 - 2k 12 X x X 2 + (k 22 + k 12 )X 2 2 

Q(K A,) = 

m^Xi + rn 22 X 2 2 
5X 2 - 6X X X 2 + 4A 2 2 

x x 2 + n 2 

As a first trial, let X x = X 2 — 1. 

5-6 + 4 

0(1, 1) = — = 0.75 

^ V ; 1 + 3 

By substituting w 2 == 0.75 into either of the equations in (3) or (5), 

we get some idea about how good our starting assumption is. In this case 

we get 

-m n co 2 + k n + k 12 = 4.25 = ^ ^ 

k 12 3 

The ratio X 2 /X 1 = 1.42 is most likely a better approximation. Rayleigh's 
quotient for this ratio is 

5-6- 1.42 + 4- 1.42 2 ^ ,^ 

W' 1A2) = . +3-1.42* =°- 64 ° 

If we take this value to be a> 2 , we have co = 0.800. 


The correct solution to three significant figures is co = 0.800 and 
Ag/Aj = 1.45. To show how close Rayleigh's quotient gives the square of 
the natural frequency when the assumed amplitude ratio is reasonably 
correct, let us observe that 

g(l, 1.3) = 0.652 co = 0.810 

Q([, 1.6) = 0.651 co = 0.809 

In other words, 10 per cent error in amplitude ratio produces about 1 per 
cent error in frequency. 

The other natural frequency can be obtained in a similar way. First we 
know that co" 2 > co^ 2 = 5. If we assume that co 2 = 5, the second equation 
of (3) gives Aj/Ag = — 1 1/3. (The first equation is too sensitive to changes 
of co 2 when it is around 5.) Next we find Q(— 11, 3) = 5.67. Repeating 
the process once, we find ^J^ 2 = —4.34 and co 2 = 5.69. These values 
are nearly the correct answers. 

In concluding this example, let us mention that the procedure demon- 
strated unfortunately cannot be adapted in a practical way to systems with 
more degrees of freedom. The purpose of this example is to show that 
when the amplitude ratio of a given mode of vibration is known approxi- 
mately Rayleigh's quotient provides the corresponding natural frequency 

(b) rayleigh's method for systems with known influence 

In contrast to the example just given, the following illustration has 
practical utility because it can be immediately adapted to systems with 
more degrees of freedom. The method to be discussed deals with systems 
in which the influence coefficients instead of the elastic constants are 
known. To apply the procedure previously shown, it would be necessary 
to find the elastic constants according to relationships such as (22). This 
is very inconvenient, especially for systems with many degrees of freedom. 
A modification of the procedure is therefore needed. 

Consider the transverse vibration of a beam supported at the ends and 
carrying two concentrated masses weighing W n and W 22 , respectively, as 
shown in Fig. 48. As usual, we start with the first trial. 

K = h 

In the previous example we determined or by Rayleigh's quotient 
corresponding to the X l = A 2 and then obtained a better estimate of the 
amplitude ratio by the equilibrium equations (3). From the revised 
amplitude ratio the frequency is then obtained from Rayleigh's quotient 
to a close degree of approximation. We now show that a revised amplitude 


ratio can actually be obtained directly without computing Rayleigh's 
quotient first. Let us write (24) as 

h = Wl7l2^1 + W 22 y 2i ?.Jo) 2 lg 


^1 = W ll7lA + F*Wl2*2 ... 

/ 2 ^Il7l2^1 + ^22722 / 2 

An iteration process can then be instituted by putting the estimated ratio 
into the right-hand side of the equation and obtaining a revised ratio from 
the left-hand side. If our assumed first-trial ratio is (/ 1: //. 2 )i = 1- tne 
revised ratio is ,' , „ r , TTr 8 

(h) = ^H7ll + ^22722 = £l (iij 

U 2 / 2 ^ n y 12 + W 22 y 22 b 2 

in which b ± and 6 2 are the static deflections of the shaft that would be 
produced by the gravity forces on the weights if the beam were in a 
horizontal position. Now we can compute Rayleigh's quotient corre- 
sponding to the revised ratio by letting k ± = b ± and / 2 = b 2 . The potential 
energy of the elastic forces corresponding to these deflections will simply 

be V m = «^iA + W 22 b 2 ) (iv) 

and the value of V m is thus obtained without the need of converting the 
influence coefficient into the elastic constants. Finally 

n(h ,, Wi + ft, Ate M 

If Aj = X 2 is anywhere near being a reasonable first-trial ratio, (A 1 /A 2 ) 2 = 
b 1 /b 2 will represent an improved estimate, and Rayleigh's quotient above 
should yield the natural frequency very closely. It must be realized that 
there is no inherent reason for AJAg = b 1 /b 2 always to be a good estimate. 
For example, if the beam in Fig. 48 had a third pivot support between 
the masses, the free vibration with the lower frequency would be such that 
the masses would move in the opposite directions. If static deflections b ± 
and b 2 are to be taken as the amplitudes, it is more reasonable to compute 
them as if the gravity forces acting on the masses were in the opposite 
directions. The results obtained when such artifice is needed are not 
generally so reliable. 

Generalization of the expression (v) to cover systems with more than 
two masses is immediate. 

WiA 2 + w^b* ■ ■ ■ w„„6 



Usually this approximation is good only for the vibration mode with 
the lowest frequency, in which the static deflections more closely approxi- 
mate the amplitudes. 

The appearance of the constant g in (ii) and (vi) does not mean that the 
gravitational force has an effect on the natural frequency. If we were to 
perform our measurements on the moon, where g is much smaller, the 
(5's would also be proportionally smaller. The resulting value of co 2 would 
still be the same. 

2.13 Some Principles in Vibration Control 

Methods and techniques in vibration control and isolation are matters 
in art as much as in science. As pointed out by Macduff and Curreri, 19 the 
problems involved are psychological, economical, and analytical, and 
their degrees of difficulty are usually in that order. Within the scope of this 
book only the basic principles concerning theory of vibration can be 
discussed. From these principles successful solutions of practical problems 
may be carried out if one also possesses experience, ingenuity, and persuas- 
iveness. The last quality is needed by the consulting engineer to convince 
the client that the methods proposed are proper or the results achieved are 

(a) removal or minimization of excitation forces 

Ideally, the way of removing vibrations in machines is to eliminate or 
minimize the sources of excitation. In practice, this is not always feasible. 
Much can be accomplished sometimes in the design stage of certain types 
of equipment. Others, such as punch presses, high-speed card printers, 
and riveting machines, have excitation sources that are inherent to their 
functioning. In an existing machine, if avoidable and excessive sources of 
excitation are suspected, measurements of the objectionable vibration 
should be taken and analyzed. If the vibratory motion is found to be 
clearly periodic, with a fundamental frequency forming a simple ratio 20 
with the rotating speed or reciprocating frequency of certain parts of the 
equipment, the source or sources of the excitation may be located and 
sometimes removed by proper balancing and replacement of faulty parts. 
When possible, efforts should also be made to vary the running speed and 
to observe whether amplitude "peaks" at a certain particular speed. 

19 See Macduff and Curreri, Vibration Control, McGraw-Hill, New York, 1958, 
p. 117. 

20 The resulting vibration can be the harmonics or the subharmonics of the excitation. 
Subharmonics are produced by the nonlinearity in the system and have frequencies 
lower than that of excitation. 


Should that be the case, the trouble lies with resonance, not with large 
unbalanced disturbances. The corrective measure is then to separate the 
resonant frequency from that of disturbing forces. The kinds of objection- 
able but avoidable vibrations described in this paragraph need not happen 
if proper design and manufacturing procedures are employed ; but they 
do happen just the same. 

(b) the lowering of natural frequencies and the effect of 

A cardinal principle of vibration control demands that the frequency of 
excitation be kept away from any of the natural frequencies of the system 
in which vibration is to be minimized. In most practical cases this principle 
also demands that the natural frequencies be kept low. In spring-mounted 
machinery springs with low stiffness will have to be used to keep the 
natural frequencies low. Soft springs, however, pose practical problems 
of their own. For instance, helical compression springs of low stiffness 
may be too unstable without lateral guides. Also, the large static deflec- 
tions associated with soft springs may also become undesirable when the 
total weight of the mounted system is a variable. For example, if the 
suspension of an automobile is too soft, the road clearance may change 
drastically as passengers enter. 21 Apart from the stiffness of the springs 
used in mounting, their placement may also affect the natural frequencies. 
Consider the system shown in Fig. 55, which may be taken as a simplified 
representation of a long narrow machine mounted on springs. Let us 
assume that the springs are guided so that the possible motion consists of a 
vertical displacement of the center of gravity y and an angular displace- 
ment 6. For small oscillations the energy expressions for the'system are 

T = \{Mf + Id 2 ) 
V = i[k(y - adf + k(y + bOf] 
= \[2ky 2 + 2k(b - a)yd + k(a 2 + b 2 )6 2 ] 

Using the notations in (17), we have 

2k . k(a 2 + b 2 ) 


11* - jjj -22 J 

CO, J = 

k(b - a) 

12 " Vm 

Evidently, if the springs are located so that a = b, there will be no coupling 

21 A desirable feature of torsion bar springs and of air springs is that the road clearance 
can be readjusted by an automatic device after the loading or unloading of the vehicle. 




between the coordinates, and the natural frequencies are co n and eo 22 . 
The difference between a and b creates a coupling, and, according to (16), 
this coupling has the effect of separating the two frequencies and making 
the higher of the two still higher. Also, for a given distance between the 
springs an increase in the difference a — b tends to make co 22 higher, since 

2(a 2 + b 2 ) = (a + bf + (a- bf 

The existence of couplings between coordinates is therefore undesirable 
in our effort to keep all of the natural frequencies low. 



k §: 

a — 



Figure 55 

In discussing the effect of couplings on the natural frequencies, a 
conceptual point must be cleared up. In Art. 2.4 it was pointed out that for 
a given system the existence and the nature of coupling are matters 
dependent upon the choice of coordinates, whereas the natural frequencies 
are properties independent of the choice of coordinate system. For 
instance, in the system just studied, \iy denotes the displacement of a point 
midway between the springs instead of the center of mass, we will find 
inertia coupling instead of elastic coupling. In the context of our present 
discussion, when we speak of the effect of couplings upon the natural 
frequencies we mean the effect of physical rearrangements that cause 
changes in the coupling constants a 12 and c 12 between a given pair of 




Generally speaking, the presence of symmetry favors the elimination 
of coupling forces. When a piece of equipment is spring-mounted at the 
base, it is usually possible to place the springs so that the vertical motion is 
a principal mode. On the other hand, as long as the center of mass is 
above the base support, any possible horizontal translation will be 
coupled to rotary motions. (See Fig. 56.) 


Figure 56 

Aside from its effect on natural frequencies, a coupling between the 
coordinates also couples the effect of external disturbances. In Fig. 55 
an up-and-down force acting on the center of mass will also produce a 
rotary motion unless the two springs are properly located. If the natural 
frequency of the rotary motion is near to that of the force, decoupling 
them from each other will also produce a beneficial effect. 


A dynamic vibration absorber is a small vibratory system coupled to 
a machine or structure to control its vibrations. The absorber is designed 
so that when the machine is subjected to a periodic excitation the resulting 
vibration of the absorber produces a coupling force that tends to cancel out 
the excitation force. 

A simple form of this arrangement is shown in Fig. 57, where M is a 
mass simulating a machine and ^is its mounting spring. The small mass m 
and the coupling spring k constitute the absorber system. A damper c 
may or may not be present. Assume that all motions are confined in the 
vertical direction. Let Y be the displacement of M and y, the relative 
displacement of m with respect to M. The energy expressions in terms of 


these coordinates are 


= i[(M + m) Y 2 + 2m Y y + mf] 
The power dissipation in the damper P is 

P = cf 


Let Q(t) be a force acting on M; the energy equation for the system is 

d , 

~(T+V) + P=QY 

From this equation and the energy expressions we obtain the differential 
equations of motion: 

(M + m)Y + KY+my= Q(t) 
mf + my + cy + ky = 


This set of differential equations can also be obtained from equilibrium 

Suppose that Q is a sinusoidal force and that we are interested in the 
steady-state response of Y. Let 

Q(t) = Fe iu) f l and Y = leT* 

According to (42) and (43), 

F(—ma) f 2 + ic(D f + k) 

1 = 


D = [-{M + m)aj f 2 + K](-mco f 2 + k + icco f ) - (nuo/f 
= (K — Mco f 2 )(k — mco f 2 + icco f ) — ma) f 2 (k + icco f ) 

Evidently, if we design the absorber in such a way that kjm = a> f 2 and 
c = 0, the amplitude of 7 will vanish. Hence, if the excitation is a 
sinusoidal force with a fixed frequency, a simple and undamped dynamic 
absorber "tuned" to that frequency will keep the vibration of the main 
structure or machine to a minimum. 

Although such simple absorbers are very effective when the excitation 
force is at the designed frequency, they are of little value if oo f is subjected 
to variations in a wide range. To prevent the amplitude of Y from 
becoming excessive at all frequencies, it is necessary to introduce damping 
in the absorber system. The following analysis for the optimum design 
of a dynamic absorber is based upon the works of a number of investi- 
gators. 22 Returning to the general solution for A, we divide both the 
numerator and the denominator by Kk and call 

F A k 2 K H2 A CC °f 

— 3= o e . — = co 2 — = £2 2 and — — = rj 

K st m M k ' 

The nondimensional equation obtained is 


+ irj 

m (co f \ 2 ,„ 

22 See Timoshenko, Vibration Problems in Engineering, Van Nostrand, Princeton, 
N.J., 3rd edition, pp. 213-220, and Den Hartog, Mechanical Vibration, McGraw-Hill, 
New York, 1956, 4th edition pp. 93-106. 



To simplify the expression, set 



m A 

- = y and 

j- = T(ico f ) 

(footnote 23) 

a + /?y 

d st (8(a + ty) - y(l - /SKI + A}) 
a + m? 

~ (a0 - y + y£) + i(P -y + ypft 

III" = ^±^ (i) 

The object is now to design a system so that \T\ will be kept low for a 
wide range of co f . Let us observe that, of all the parameters determining 
|j|, tj is the only one containing the damping constant c, and that when 

( g L p-y + yp \ 
a =±1-5 t (n) 

\T\ is independent of r\. Physically, this means that for certain combina- 
tions of m, k, M, K, and co f the amplitude of vibration of M may be made 
independent of the damping constant c. To determine such combinations 
more specifically, we find that 

= «g - 7 + yP 
P-y + yp 

is satisfied by a = 1 , p = 1 , or y = 0. These values imply the trivial 
conditions m = or co f = 0. The other possibility 

x= _ «e-jy + yj} 

P - y + yfi 

leads to 

(2 + y)aj8 - (a - p + l)y = 

which expands into 

1 + -W - (^ 2 + co 2 + yco 2 )u f 2 + w 2 Q 2 = 


This is a quadratic equation of co f 2 . It can be shown that the discriminant 
of this equation is 

(Q 2 - (o 2 ) 2 + yo/(2 + y) > 

23 There is no danger of confusing the symbol for transfer function with that for the 
kinetic energy of the system. So we use T for both. 




and the equation has two positive roots for oj f 2 . In other words, if a 
family of frequency response curves (\T\ versus oj f ) is plotted with damping 
constant c as a parameter, all the curves will have two points in common, 
as shown in Fig. 58. 


7 = 0.05, g = 1 
c x = 0.64 Wmk, c 2 = 0.20Vmfc 

Figure 58 

The abscissae of these two points, to x and co 2 , are determined by (iv), 
and the corresponding ordinates |7\| and \T 2 \ are determined by (i) and 
(iii). They are 

1 7i| = —T-, — and |rj = — ; — (v) 


±(0i - y + rA) 


T ^ T(0 2 - v + yh) 


e) : 

The signs to choose are the ones that make |T|'s positive. It can be shown 
that the correct signs are such that one is positive while the other is 

To achieve an optimum design of the vibration-absorber system our 
approach is now as follows. We consider M and K as being fixed because 


they belong to the system whose vibration is to be controlled. Tentatively, 
we try to utilize the freedom we have in choosing three parameters, m, 
k, and c, to achieve the following three things. One is to make IrJ = |T 2 |, 
and the other two are to make |7\| and \T 2 \ the maxima of the frequency 
response curve. It is reasonable to expect that an optimum design would 
be achieved when the ordinates of the two points, through which all 
curves must pass, are made to be the two maxima having equal values. 
Upon setting 

\n = n 

we have 

(A - v + yA) = -(& - y + yft) 

Substitute in the expression for p x and /? 2 and simplify. 

2H 2 
V + co 2 2 = — - — (vi) 

1 + y 

Since co^ and co 2 2 are roots of (iv), their sum is 

20 2 2(H 2 + co 2 + yw 2 ) 

This simplifies into 


1+y 2 + 7 




1 + 7 ( vni ) 

m K / ra V 
Affc \ + MI 

This is one relationship that must be satisfied in an optimum design. The 
corresponding value of \T\ is then 

\t\ = |rj = |r,| = K|rj + |r 2 |) 

Substituting (v) into the above, we obtain through simplification 

ii ii 2coQ, 

\ T i\ = Fa = 7—2 ^ lf w i > w 2 

This expression evidently has to do with the difference between the 
roots of the quadratic equation (iv), which may be written as 

/coiy _ 2[Q 2 + co 2 (l + 7)] /oy>\ 2 _ Q 

Uq; " (2 + 7)wft Uq/ 2 + 7 

By utilizing (viii), we have 


Treated as an equation in co^/coH, the difference between its roots is 

2 2 - 4 J^— = 

2 + y N 2 + y 

Z±l = J l+ ™ ( X ) 

y 'V m 


con ± |r| 

To make |r| a maximum at o, = co x and coy, = od 2 , we demand, if 
possible, at oj f = a^ and co,- = co 2 

d(co f 2 ) 
For simplicity let us write (i) as 

\T\ 2 (A 2 + #V) = a 2 + ?f 

in which A = a/5 — y + y/3 and B = j3 — y + yfi, and use the symbol 
prime (') to denote differentiation with respect to co f 2 . The condition (xi) 
leads to 

\T\ 2 (AA' + BB'rj 2 + £ 2 ??V) = oca' + ???/ 

According to (v) and (iii), at co-^ and co 2 

|r| 2 5 2 =l and ^ = a 


#V = £aa' + v.A' 

By utilizing the definitions of A and 5 and the relation 

j8' co 2 V ^ 7; 
we obtain through some simplification 

(1 + y)rj 2 = a[(l - y)(B + p) + a + y] (xii) 

Now the solution of (ix), together with (v), (viii), and (x), gives 

% = d-B) % = (i - m + r) 

coll OT 

< = (Ll9 and a = ±7^ 
Q 2 (1 + y) ^ r + 2 





Optimum conditions 

#=1+7, 7 = 0.2 

c 2 _ 3yk 2 
optimum — 2 uQ 


" 1 



1 ° 















Figure 59 



Elimination of a, ft, B, and r\ in (xii) by the use of (xiii) produces the final 


2 W Q = 3y ± yVyl(2 + 7) 


in which the "+" sign is for co f = c^ and the " — " sign for co f = co 2 with 
a) x > co 2 . Since 7 ^ 0, it is not possible to make \T\ a maximum at both 


(0-l and co 2 . Experience has shown, however, that if T versus the m t curve 
reaches a maximum at one of the two co's, the other maximum has only a 
slightly higher value. Hence a compromise can be made by taking the 
average of the right-hand side in (xiv). (See Fig. 59.) 



. 3 mk 2 3m 2 koj 3 m 2 k 
c i _ 

2 MojQ. ~ 2 MQ ~ 2 m + M 
2.14 Effects of Rotation on Critical Speeds of Shafts 

In Art. 2.10 the critical speed or speeds of a rotating shaft were con- 
sidered as nothing more than the resonant frequency or frequencies of the 
system in transverse vibrations. It was maintained that the rotation 
merely furnished the necessary excitation through the existence of 
unavoidable eccentricity in the mass distribution of the system. In other 
words, if we want to determine experimentally the critical speeds of a 
certain shaft carrying one or more disks, we can, instead of rotating it, 
excite it with sinusoidal transverse forces of different frequencies and 
observe the speeds when resonant transverse vibrations take place. 

The situation represented is a simplification that serves practical 
purposes well in most cases. However, there are circumstances in which 
the fact that the shaft is rotating has a bearing on the problem. In this 
article we discuss briefly some of the effects produced by the rotation 

(a) gyroscopic effect 

In Art. 2.10 the inertia of the disk carried by the shaft is represented by 
that of a single mass particle. This is permissible if the motion of the 
disk during a transverse vibration is a pure translation. In general, the 
motion of the disk also includes a rotation about an axis perpendicular to 
the plane of bending, as shown in Fig. 60. The inertia of the disk thus 
exerts a moment on the shaft when the system is in simple transverse 
vibration. If the shaft is also rotating about its own axis, there is an 
additional gyroscopic moment as the rotating disk changes its spatial 

Let us first investigate the case of a nonrotating shaft. Assume that the 
system vibrates in the xy-plane. Let y and d denote, respectively, the 
deflection and the rotation of the cross section of the shaft where the 
disk is located. During a transverse vibration in the xy-plane the disk 
exerts on the shaft a force my and a moment Ifi. When the system is in 


simple harmonic vibration, y and 6 may be represented by 

y = X cos cot 6 = j3 cos cot 

At the extreme position of the disk, the inertia force and moment are as 
shown in Fig. 60. 

Figure 60 

We now define the influence coefficients: 
y f — deflection of a cross section per unit force applied at the cross 

y m = deflection of a cross section per unit moment applied at the cross 

(f) f = rotation of a cross section per unit force applied at the cross 

(f) m = rotation of a cross section per unit moment applied at the cross 
Maxwell's reciprocal relation is also valid here, so that y m = <j> f . With the 
help of these coefficients, we can relate the deformation of the shaft with 
the force and moment produced by the inertia of the disk: 

X = moj 2 Xy f + I z co 2 Py v 
p = mco 2 ?4 f + i>W, 



This set of equations is evidently in the standard form of (24). The 
natural frequency of the system is thus given by 

m 7f i hYm 


m 7m W* \ 


It can be shown by simple algebra that this frequency equation has two 
positive roots, oj x 2 and w 2 2 , and that co^ < co 2 < w 2 2 where co 2 = 
\jmy t = k\m. 

The transverse vibration of a nonrotating shaft with one disk thus has 
two degrees of freedom in the x?/-plane and another two degrees of freedom 
in the xz-plane, which are identical to those in the z2/-plane. 

Now, if the shaft is rotating, the system has five degrees of freedom. 
The natural frequency in the fifth degree is zero, since it is a rigid-body 
rotation. However, because of the gyroscopic effect a nonlinear coupling 
exists between the rotation about the a>axis with that about the y- and z- 
axes. A general analysis of the problem requires Euler's equation for the 
rotation of rigid bodies. There is, however, a simple and practical situation 
that we can analyze with elementary considerations. 

Let us recall that the essence of the result obtained in Art. 2.10, as 
expressed by (58), is that at co f 2 = kjm a shaft carrying a mass particle 
with zero eccentricity is in a state of neutral dynamic equilibrium. It 
can rotate in a bent shape as well as in a straight shape. For a real shaft 
with a small but unavoidable eccentricity this speed marks a condition of 
instability. The question is now whether the situation is still the same when 
the mass carried by the shaft is in the form of a disk. Let us suppose 
that the shaft and the disk rotate together around the x-axis in a bent 
shape, as shown in Fig. 61. The centrifugal force acting on each mass 
element of the disk is proportional to the displacement of that element from 
the x-axis. This distributed centrifugal force produces a resultant force 
perpendicular to the a>axis and a resultant moment which tends to restore 
the disk to its neutral position. A comparison of Fig. 61 with Fig. 60 
reveals that the moments of the inertia forces have opposite directions 
when the system is in transverse vibration and when it is rotating around 
the x-axis in a bent shape. 

To determine the force Fand the moment M in Fig. 61, we can integrate 
the contributions of each mass element of the disk with the help of an 
appropriate coordinate system. The procedure is routine but a little 
tedious; therefore we shall analyze the problem by a different approach. 

The motion of the disk is assumed to be made up of a whirling motion, 




which is a translation, and a rotation about an axis parallel to the x-axis. 
The translation produces a linear momentum whose time rate of change 
results in a centrifugal force F given by 

F = moo f 2 k 

The rotation produces a moment of momentum L whose time rate of 
change results in a moment M to be found as follows. Let us first observe 

Figure 61 

that the angular velocity vector co f , being along the a>axis, does not coincide 
with any of the axes of principal moment of inertia of the disk. Therefore 
the moment of momentum vector L is not in the direction of co f . (See 
Fig. 62.) Let I x denote the principal moment of inertia of the disk about 
its centroidal axis perpendicular to the plane of the disk. This axis is also 
tangent to the deflection curve of the shaft. The component of L along this 
axis, called L l5 is 

L x = I^Wf cos ft) 

Let I 2 denote the principal moment of inertia of the disk about the dia- 
metrical axis that lies in the plane of bending. The component of L along 
this axis is 

L 2 = I 2 (a> f sin fl) 


The z-component of L is therefore 

L x = L x cos p + L 2 sin ft 
= (I t cos 2 + I 2 sin 2 ^cw, 


= ^ + w, cos2 ^ W/ = 4w/ 




L ___ — 


L 2 = I 2 oo f sin /3 

Figure 62 

and the ^/-component of L is 

L y = L x sin ($ — L 2 cos 

= ( 1 2 2 sin 20 j w / = /„©, 

in which 4 is the moment of inertia about the a>axis and I xy is the product 
of inertia in the a^-plane. 

The so-called ^-component, however, has no fixed direction, since it 
always lies in the plane of bending and rotates with it. The time rate of 
change of L is the same as that of L y and is represented by a vector in the 
z-direction of a magnitude given by 

dL v T T 2 

— = oj f L y = I xy aj f 2 

According to d'Alembert's principle, the moment produced by this change 
is represented by a vector in the opposite direction, which gives the direc- 
tion shown in Fig. 61 through the right-hand rule. 
When f$ is small, this inertia moment becomes 

M = 


- -J> f = -ft - hWP 




The minus sign indicates the fact that M is opposite /?. Using the influence 
coefficient previously defined, we have 

X = mo) f 2 ky f - (/i - I 2 )w f 2 py m 

P = mco f ny m - (I ± - I 2 )a> f 2 p(f> m 


0)f COS 

L\-I x (co f cos /3 - 2w f ) 

L 2 -I 2 u)f sin /3 

Figure 63 

Since this is a set of linear homogeneous equations in A and ft it admits 
nonzero solutions only if 

mv '~h 


~(h - h)Ym 

co i 



For circular disks whose axial dimensions are small in comparison with 
radial dimensions, we have 

h-h = h 


h = 


my f 







In comparing (v) with (ii), we see that the terms containing I z have different 
signs. As a result, there are two positive roots for (ii) but only one such 
root for (v). Hence for a shaft carrying a single disk there is only one 
critical speed at which the system is in a state of neutral dynamic equili- 
brium. (See Exercise 2. 1 3.) 

It has been observed in practice that sometimes the gyroscopic effect can 
cause large vibrations in a rotating shaft at another speed, at which the 
shaft whirls in one direction while the disk rotates in the opposite direction 
at the same speed. A little reflection will reveal that the rotary motion of 
the disk in this case can be represented by the sum of the two angular 
velocity vectors, shown in Fig. 63. The ^/-component of the moment of 


momentum vector is then 

L y = I^—lco f + (o f cos p) sin p — I 2 w f sin p cos ft 
and for small p 


Figure 64 

The moment due to the inertia force is thus 

M = (Ji + 7 2 )^ W/ 2 

This time the moment is in the same direction as p. The rest of the pro- 
cedure is the same as before, and we obtain 

™Vt 5 (A + / 2 )y, 





This equation has two positive roots for co f 2 . The solutions to (iv) and (vi) 
can be put into a common nondimensional form: 

(S) 2 = \ [ ° + aIf ± V(l + a/) 2 + 4te/] (vii) 


in which co 2 = \\mr f is the critical speed ignoring the gyroscopic effect 

^™ JL 2 

a = = 0m°>O 


v 2 
6 = i __^_ (0<6< 1) 

/ is the appropriate moment of inertia appearing in (iv) or (vi). The 
hyperbola shown in Fig. 64 describes the relationship between c» f and /. 
The relative magnitudes of the different roots of (iv) and (vi) are as 

Let us call the velocity condition described by Fig. 62 a forward whirling 
and that by Fig. 63 a backward whirling. Figure 64 shows that forward 
whirling can take place only at one speed, whereas the backward whirling 
can take place at two speeds. It was assumed that in either forward 
whirling or backward whirling the whirling speed is numerically equal to 
the rotation speed. A question that may be asked is then, "Can the shaft 
whirl and rotate at two different speeds?" Reviewing the method of 
analysis employed, we see that given an arbitrary ratio of the whirling 
speed to the rotation speed we can find, at least theoretically, the speed or 
speeds of rotation at which the assumed ratio is realized. Since that is the 
case, we must ask, "Why are the speeds given by (iv) and (vi) called 
critical?" The answer to this question lies within the fact that all the 
motions postulated are free motions in the sense that there is no energy 
input. Because of the unavoidable dissipative forces in real systems, such 
motions cannot be sustained without some kind of suitable excitation. 
In the case of forward whirling at the speed of rotation the needed excita- 
tion is immediately available in the form of a centrifugal force produced by 
the unavoidable eccentricity. The critical speed given by (iv) is therefore 
always observable in practice, and it may be considered as the true critical 
speed. The backward whirling cannot be excited as effectively by the 
eccentricity, but a real shaft, which is a part of a piece of machinery, may 
be subjected to other suitable excitation created by the rotation. Backward 
whirling is therefore only occasionally observed. 24 Motions in which the 
whirling speed and rotation speed are entirely different do not usually 
have the necessary excitation to sustain them. 

(b) effects of gravity force 

If a shaft carrying a disk is in a horizontal position, the gravity force 
will cause it to assume a bent shape even when it is at rest. But aside from 

24 See Den Hartog, op cit., p. 265. 




producing this change in the equilibrium configuration of the system, 
the gravity force has no effect on the frequency of the free vibrations of a 
nonrotating shaft. The situation, however, is different when the shaft 
rotates in a horizontal position. 

Consider the simple case in which the gyroscopic effect is not present. 
During rotation there are three forces acting on the disk: a gravity force, 
an inertia force, and an elastic restoring force, as shown in Fig. 65. In 

Figure 65 

this figure is the neutral position of the center of the shaft when it is 
straight and 6 is the angle between the horizontal plane and the plane 
of bending of the shaft. A normal way of operation is obviously that in 
which the plane of bending remains vertical so that the gravity force 
balances the elastic force and there is no inertia force as the disk rotates 
about its lowered center M. The question to be asked now is, "Is it also 
possible to achieve a dynamic equilibrium with the plane of bending 
whirling around at a constant speed co f T Assume that this is possible. 
Let us first resolve the inertia force into a radial component nuo f 2 r — mr 
and a tangential component 2mco f f, which is the Coriolis force. The 
whirling speed co f is assumed to be constant, and d = {co f t — a). Hence 

mr + (k — mco f 2 )r = — mg sin (co f t — a) 


2mf(o f = — mg cos (co f t — a) 

Obviously, these equations are satisfied by co f = 0, a = 77/2, and r = 
constant = mgjk. Another way of satisfying (viii) is for co f to have the 



appropriate value so that 

r = r sin (co f t — a) (ix) 

To see that this is true we substitute (ix) into (viii), and obtain 

—mco f 2 r + (k — mw f 2 )r = —mg 

2mco f 2 r = — mg 

(k — 4mco f 2 )r = 
oj f = \Vkjm = \coq 


where = co f t — a, the path traced by M as the plane of bending whirls is 
a circle having radius d st and lying below but tangent to the horizontal 
plane containing 0. 25 

Theoretically, the motion described can take place regardless of the 
speed of rotation of the disk. But again without suitable excitation this 
motion cannot be sustained in real systems with the ever-present damping 
forces. Tn real shafts the unavoidable eccentricity causes a whirling motion 
at the speed of rotation. When this speed is cojl, the motion produced 
by the gravity force reinforces that produced by the eccentricity, and the 
resulting motion is represented by 

• • x (0 o 2e 
r = r sin (co f t - a) + — 



— — 

-2S 8t 

2mco f 2 




r cos 

= r sin 


= — 

&8t Sil1 




r sin 

= r sin 2 




— COS 


. o)J \ 4e 
— o af sin , 

1 2 

With or without eccentricity, at the so-called critical speed of the second 
order, co f = co /2, the vibrations produced have only finite amplitudes, 
which are of the order of d st . Sometimes vibrations at this speed are 
observed in practice with a substantially larger amplitude. Such vibrations 
are usually due to other causes, such as a difference in the rigidity of the 
shaft and its support in two different planes. 

25 This motion is the same as one of those described in Exercise 1.35. It is, however, 
instructive to compare the difference as well as the similarity between the system 
described in Exercise 1.35 and the one being studied. 





It has been observed in practice that a shaft rotating above its critical 
speed may develop a whirling motion that is near its critical speed. This 
phenomenon is commonly attributed to the existence of a hysteresis loop 
in the shaft material. To explain such whirling motion on the basis of the 
hysteresis characteristic of the material, we begin with a simpler pheno- 
menon with which the structural damping law of Kimball and Lovell, cited 
in Art. 1.18, was first deduced. Consider a rotating shaft supported 
between two bearings and subjected to an external lateral force P, which 
causes the shaft to bend as shown in Fig. 66. If the shaft material is 
perfectly elastic, the plane of bending will contain the force P, which 

Figure 66 

causes the bending. But if the shaft material has a hysteresis loop in its 
stress-strain relation, as shown in Fig. 61a, the plane of bending will not 
contain P. To arrive at this conclusion, let us assume that the plane of 
bending is the f/z-plane in Fig. 66. Consider a cross section of the shaft, 
which is in the xy-p\ane. The fibers above the xz-plane, such as those 
represented by points A and B in Fig. 61a, are under tensile strains, 
whereas the fibers below (C and D) are under compressive strains. As 
the shaft rotates in the direction indicated, it is not difficult to see that A 
and C are being loaded, whereas B and D are being unloaded. Therefore, 
if Fig. 61a represents the stress-strain history of all the outer fibers of the 
rotating shaft, the stress-strain states of fibers A, B, C, and D are given by 
those correspondingly labeled points in Fig. 67a. Hence the tensile stress 
is higher at A than that at B, and the compressive stress is higher at C 
than at D. If we consider all the fibers in the shaft, the net result is an 
angle between the neutral axis of zero strain x — x and that of zero stress 
n — n, as shown in Fig. 61b. Hence the external force P required to keep 
the plane of bending in the ?/z-plane must be making an angle a with the 
2/-axis, as shown. Let the deflection of the shaft where P is applied be 




|A|. The x-component of the force P and those of the reactions at the 
bearing form a torque which must be balanced by an external driving 
torque in order to maintain the speed oj f and the deflection |A|. The energy 
input per revolution of the shaft is therefore 

E* = 2ttPM\ = 2ttP sin a 


This energy is dissipated by the hysteresis of the material. 

As discussed in Art. 1.18, the experiments of Kimball and Lovell 
established that 

E t oc \Xf 

Hence P oc |A|, and the angle a is a measure of the structural or hysteresis 


Figure 67 

damping force in the system. To summarize, if a rotating shaft is deflected 
in a certain direction, it offers a resisting force which has two components. 
One component is due to the elasticity of the shaft, and it is proportional 
and opposite to the deflection. The other component is due to the 
hysteresis of the stress-strain cycling suffered by the material in a rotating 
shaft, and this component is proportional and perpendicular to the 
deflection. It is 90° from the elastic component measured against the 
direction of the shaft rotation. 

As pointed out in Art. 1.18, strictly speaking, the description of the 
hysteresis force given above holds true only if the deflection of a rotating 
shaft is static. It is, however, reasonable to assume that this description 
also holds when the deflection varies slowly with respect to time; that is, 
the bending motion is slow in comparison with the rotation. Let us now 
consider a shaft carrying an inertia mass and rotating at a speed above its 
critical. In the absence of other forces the equilibrium configuration of the 




system is such that the shaft is straight. Suppose that a transient distur- 
bance causes the mass to move away from the center-line of rotation. 
Now we want to study the motion that ensues. The restoring force and the 
inertia force on the mass are shown in Fig. 68. Let k and h be the constants 
of proportionality of the elastic force and the hysteresis force, respectively. 
The differential equations of motion are therefore 

mx + kr cos 6 — hr sin 6 = 
my + kr sin 6 + hr cos 6 = 


Center-line of 
undeflected shaft 

Figure 68 

Since r cos 6 = x and r sin 6 = y, 

mx + kx — hy = 
my + ky + hx = 

If we let £ = x + iy, the two equations can be combined to yield 

ml + (k + /'/OS = 

In the Argand diagram the complex variable £ graphically describes the 
location of the mass m. Defining as usual h = i]k with i] = tan a > 0, 
we have finally 

ml + k{\ + irfrl = 
or (hi) 

ml + k£ = 

where ac = &(1 + irj). The solution to (iii) is 

I = 4^ + Be*** 




1 = ±icoV\ + it} (v) 

and (o = Vkjm = critical speed in the absence of hysteresis. 
Through the usual complex number algorism, we obtain 

S 2 \ 

For small ?] or ?/ <^ 1 

i + Vi + v 2 ± / l + Vi +t i ) co 

2 J 2 

VI + yf = 1 + hf 



By substituting the expressions for s 1 and 5 2 into (iv) and remembering that 
^ describes graphically in Argand's diagram the location of the mass in the 
£?/-plane, we conclude that the motion of m consists of a counterclockwise 
whirling (+ico) with decaying deflection ( — fja>j2) and a clockwise whirling 
( — ico) with increasing deflection (+rjcoj2). The latter motion represents 
an instability of the system. Since the rotation of the shaft viewed from the 
same plane is clockwise, the unstable whirling is with the rotation. 

Returning to the physical picture, we observe that the foregoing con- 
clusions are contingent upon these assumptions: 

(1) o) f > co; otherwise the experimental law (i) cannot be used. 

(2) There are external driving torques sufficient to maintain the speed 
of rotation m f . 

(3) There are no other damping forces operating. 

In an actual case a viscous force, such as air damping, will limit the 
whirling motion to small amplitudes. 

To close this discussion, let us mention one mathematical observation. 
In Fig. 68, if we reverse the direction of rotation co f , the direction of the 
force hr will also be reversed. Consequently (ii) and (iii) will be modified 

mx + kx + /?«/ = 

m V + ky — hx — 

ml + Rt = (vi) 

where R = k(l — irj) is the complex conjugate of k. Since (vi) can be 
written as 

ml + kZ = 


Its solution is complex-conjugate to that of (iii). This time the unstable 
whirling is in the counterclockwise direction, which is still the direction 
of shaft rotation. Hence the physical picture remains the same. In other 
words, whether we call k == k(\ + irj) the complex stiffness or R = 
k{\ — irj), the result is the same. 

(d) oil whip 

When a shaft is supported by journal bearings with hydrodynamic 
lubrications, the lift force produced by the oil film may pulsate and 
interact with the elastic and inertia forces of the system to produce large 
vibration. A detailed analysis of the problem is quite complicated. As a 
first approximation, we may neglect the elastic effect and consider the shaft 
to be perfectly rigid and straight. We know from hydrodynamic theory 
of lubrication that in a journal bearing with a certain amount of clearance, 
the journal is displaced from the concentric position, a lift force is pro- 
duced by the oil pressure distribution. This lift force is perpendicular in 
direction and proportional in magnitude to the displacement of the 
journal. 26 If a shaft is running with no transverse load and its journal is 
displaced from the concentric position by a transient disturbance, this lift 
force may cause the journal, hence the shaft, to whirl. The situation is 
similar to the hysteresis effect already discussed. The equation of motion 
may be written 

mx -f hy = 

my — hx = 

where h is found by Robertson 27 to be 

h = H((o f - 20) 

with H a constant determined by the bearing dimensions and lubricant 
property, co fi the rotation speed, and d, the angular velocity of whirling. 
This set of differential equations can be satisfied by 

6 = constant = co 1 

X ) m t f C0S / 

y) Urn 


2ma) 1 2 — H(co f — 2co x ) = 

In bearings of usual design H is often a large number in comparison with 

26 If the displacement is small in comparison with the clearance. See Norton, Lubri- 
cation, McGraw-Hill, New York, 1942, pp. 106-113. 

27 "The Whirling of Shafts," by D. Robertson, The Engineer, Vol. 158, 1934. 




mco^ so that the whirling speed co x is very nearly equal to one half the 
rotation speed co f : 

Again we have a situation in which the displacement tends to infinity if not 
checked by other forces unaccounted for here. 

It may be noted that the two phenomena discussed in (c) and (d) do not 
by themselves favor a particular speed or speeds to take place. Based upon 
the much simplified analysis given, they will take place theoretically at all 
speeds. In practice, other contributing factors determine whether or not 
they will appear at all. 


2.1. Show that the solutions for co 2 in (4) are always positive. 

2.2. In Fig. 44, if we define x x as positive when m lx moves to the right and x 2 
as positive when m 22 moves to the left, what changes if any will we have to make 
in (1), (3), (4), and (5) ? How will the values of co and /u be affected by this change 
in sign convention ? 

2.3. In general, are x ± (t) and x 2 (t), as described by (7), periodic functions? 
Under what condition will they be periodic ? 

2.4. Substitute (8) into (1), utilize (5), and compare the results with (10). 

2.5. In Fig. 44 let m n and ra 22 weigh 2 lb each, k lt = 3 lb per in., k 12 = 2 lb 
per in., and k 22 = 1 lb per in. Let the initial conditions of the masses be 
2^(0) =2 in., x 2 (0) = 0, and ^(0) = x 2 (0) = 0. Find the motions of the two 

2.6. A mass particle constrained to move in the x^-plane is connected to a 
stationary frame by three springs of equal natural lengths as shown. 



Exercise 2.6 










(a) With x and y indicated in the figure as the coordinates, write the differential 
equations of motion for small oscillations of m and solve the frequency 

(b) Without solving the problem, we know the two principal modes of vibration 
are two mutually perpendicular rectilinear simple harmonic motions. How 
do we know this is the case ? 

(c) Find the directions of the two motions representing the principal modes. 

(d) If we use Rayleigh's method to determine the frequencies, we shall need 
a reasonable estimate of these two directions. The mode having the lower 
frequency must favor the softer spring, and vice versa. So let us assume 
that the low-frequency mode is perpendicular to the spring 3k. Find the 
frequency by Rayleigh's method and compare the result with those from 

2.7. Find the frequency equations for small oscillations of the systems shown. 

2.8. Find by Mohr circle construction the natural frequencies and the corre- 
sponding amplitude ratios of the two modes of free vibration of the system shown 
in Fig. 53, if 6 = 30°, Mg = Img = 10 lb, K = 3k =15 lb/in. 

2.9. Find and solve the frequency equation for the transverse vibration of a 
cantilever beam carrying a mass m at its free end and another mass 2m at its 
midspan. The length of the beam is L, and its section stiffness is EI. 


Exercise 2.9 

2.10. If at the free end of the beam a spring of constant k = EI/L 3 is also 
attached, find its frequency equation. 

2.11. Prove (22) by the following considerations: 

v = WA + /W 

V = JfaA" + 2c 12 d^2 + c 22 S 2 2 ) 

in which d 1 and S 2 are the deflections at two points of an elastic system produced 
by two loads P 1 and P 2 . Or 

8 i = Pi/n + P2V12 

d 2 = P\y\2 + JV22 

2.12. Show that if the potential energy expression V is always positive for 
arbitrary b x and 5 2 , which do not vanish simultaneously, it is necessary that 

> and c xl c 22 > c 12 







-yjmk ^2k 

K a >+* a 




Exercise 2.14 




V = «( AA ± Vc 22 d 2 f T 2(V Cll c 22 - c 12 )V 2 ] 

V > for d 1 = 
K > for d 2 = 

K > for V^j ± V^<5 2 = 

2.13. Show that there is only one positive solution to (iv) and (v) of Art. 
2.14, since c n c 22 > c 12 2 implies y n y 2 2 > >"i2 2 - 

2.14. Set up the differential equations of motion in terms of the coordinates 
indicated for the free vibrations of the damped systems shown and determine their 
characteristic equations. 

2.15. In the accompanying figure M represents an instrument which is packed 
in a box B with springy padding having an equivalent spring constant K. Inside 




pagssgss ^ 




Exercise 2.15 

the instrument is a small part m shock-mounted by spring k. The box with its 
content is dropped from a height h. If 

M weighs 25 lb \\K = 0.02 in/lb 
m weighs 0.5 lb \jk = 0.5 in/lb 
h = 5 ft 

Find the maximum spring force in k. 

2.16. An electric motor is mounted at the center of a horizontal beam that is 
supported at its ends by springs and dampers and is presented from sideways 



movement. An eccentric mass is attached to the rotor of the motor and is being 
turned around in a vertical plane with a constant speed to f . Let 

m = the combined mass of the assembly consisting of the rotor and the mass 

e = the eccentricity of the center of mass of the assembly 
M = the combined mass of the assembly consisting of the beam and the stator 
r = the radius of gyration of the assembly above 


M-2m I = Mr 2 r = Ae a = 2r 
Exercise 2.16 

Write the differential equations of motion for the up-and-down and rotary 
motions of the beam and find the steady-state solution. 

Hint. Replace the rotating part of the system by a vertical inertia force, a 
centrifugal force, a gravity force, and a torque. 

2.17. Find an expression for the amplitude of the relative displacement x in 
the steady-state operation of a dynamic absorber of optimum design in which 


= 1 + 






2.18. Design a dynamic absorber under the following conditions: 
M = 500 lb 
K = 100 lb/in. 

F = amplitude of disturbing force = 20 lb 
m < 50 lb 

D = coil diameter of the coil spring used for absorber spring 
D < 4 in 
S = shear stress in absorber spring S < 40,000 psi 


(a) Use the static deflection curve of the beam described in Exercise 2.9 to 
find the lower natural frequency by Rayleigh's method. 

(b) Do the same for the beam in Exercise 2.10. 

2.20. A steel shaft 1 in. in diameter overhangs a long bearing by 4 in. It 
carries at its free end a thin steel disk weighing 10 lb and having a radius of 
gyration of 4 in. Find the critical speed of the shaft, taking into account the 
gyroscopic effect due to forward whirling. 



2.21. In the system shown let the imput signal be the motion (displacement) 
of the point P and the relative displacement between the two masses be con- 
sidered as the response : 

(a) Find the transfer function. 

(b) Find the impulse response. 



c = ^Jmk 
Exercise 2.21 



(a) A simple pendulum consisting of a mass m and a light rigid rod of length L 
is attached to the end of a vertical rod by a pin joint which allows rotation 
in only one plane. If the vertical rod is spinning with a constant speed co f , 
describe the small-oscillation behavior of the pendulum. 

(b) If the pin joint is replaced by a universal joint, analyze the same. 


Exercise 2.22 


Systems with Several 
Degrees of Freedom 


3.0 Introduction 

Having studied systems with two degrees of freedom, we can now 
generalize our analysis to cover systems with any finite number of degrees 
of freedom merely by incorporating additional coordinates into our 
mathematical operation. With proper minor modifications, all the results 
obtained in Chapter 2 can be applied to systems with several degrees of 
freedom. There is, however, much to be gained by re-examining our 
problem in its more general formulation and analyzing it with more 
powerful mathematical tools. 

In this chapter we shall write our equations in matrix form. Although 
matrices by themselves can be considered mathematical entities, in 
application to our problems we can consider them merely a kind of 
shorthand. To help those readers to whom matrix notation is new, all 
the essentials of matrix algebra needed for our purpose are outlined in the 
Appendix. In the text we shall write our equations first in both the 
"longhand" and "shorthand" notations and gradually turn to writing 
in the shorthand notation only. 

A few words are needed to defend the use of this mathematical tool, 
which is not indispensable for solving practical problems. Three justifi- 
cations may be given. The first is that a simplified symbolism often provides 
a clearer picture of the physical theory it represents ; a new formulation 
may reveal new ways of solving old problems. For instance, numerical 



multiplication or division can be carried out with either Roman or Arabic 
numerals, but the Arabic are not only more manageable, they also reveal 
more about the nature of the arithmetic processes themselves. The second 
justification is that matrix algebra is the unifying language of many 
seemingly unrelated subjects. Its use enables us to borrow the experience 
and feeling gained in analyzing one type of problem for the purpose of 
analyzing another. And the third is that matrix notation has become 
the accepted language for communication on the subject. Many technical 
papers which deal with the topics discussed in this chapter are written in 
this mathematical language. Furthermore, since the advent of electronic 
computers, it has become increasingly necessary for engineers to communi- 
cate with the people who run the computers or to "communicate" directly 
with the computers themselves. Both the operators and the computers 
can understand and help to solve an engineering problem readily when 
it is presented in matrix language. 

3.1 Generalized Coordinates, Constraints, 
and Degrees of Freedom 

We begin our study with a more thorough discussion of the terms 
mentioned in this heading. Any set of quantities, q v q 2 , . . . , q n , can be 
considered as a set of generalized coordinates of a system if together with 
a time t they can be made to describe the configuration of the system. 
That is to say, when their values at any time t are known, the Cartesian 
coordinates of every mass particle in the system are determined. These 
generalized coordinates are said to be kinematically independent if the 
geometry of the system does not impose any interdependence relationships 
among them and their derivatives. In other words, it is possible for the 
system to have any set of values of q's and q's at any time t. On the other 
hand, if relationships such as 

K(q v q 2 , • • • , q n ; q lf q 2 , • • • , q n ; t) = 

exist, they are called the equations of constraint, or simple constraints of 
the system. The number of degrees of freedom of a system is the number 
of generalized coordinates, less the number of constraints among the 
coordinates. There are two types of constraints. One type is called the 
integrable constraint for which the function K contains no q's, or the q's 
can be eliminated by integration with respect to /. For example, 

K(q l9 q 2 ) = ft + 2? 2 = 

K(q l9 q 2 , fa fa) = <7i?2 + q x q 2 = 




With such constraints, some of the coordinates chosen beforehand can be 
eliminated. If all the constraints of a system are integrable, it becomes 
possible to have a set of generalized coordinates whose number is the 
same as the degrees of freedom of the system. Such a system is called a 
holonomic system and is the only type of system considered in this book. 
As an example of a nonholonomic system, that is, a system with non- 
integrable constraints, consider the motion of a coin of radius r, which can 
roll and spin without slipping on a rough horizontal table top. If it is 


Figure 69 

assumed that the coin plane remains perpendicular to the table top. the 
configuration can be specified by the Cartesian coordinates of the coin 
center, x and y, the bearing angle a of the coin plane, and the inclination 
P of a diameter fixed in the coin, as shown in Fig. 69. The nonintegrable 
constraint relationships among x, y, a, and ft are 

rp cos a = x 

rp sin a = y 

This system therefore has only two degrees of freedom but needs at least 
four generalized coordinates to specify its configuration. 

3.2 Energy Expressions in Generalized Coordinates 
for Linear Systems 

We have seen in Chapter 2 that the vibratory motion of a system is 
governed by a set of linear differential equations, if its potential energy is a 
quadratic function of its coordinates and its kinetic energy is a quadratic 


function of the first derivatives of the coordinates. This is also true for 
linear systems with more than two degrees of freedom. 

T = K«ii?i 2 + a 22 q 2 2 + • • • + a nn q n 2 • • • 

+ 2tf 12 ^ 2 + ^13^3 + 2tf 23 M3 + • • ') (1) 

y = i( c n^i 2 + c 22^2 2 + * * * + c nn q n 2 • • • 

+ 2c 12 ?i? 2 + 2c l3?l?3 + 2c 23^2^3 + " ') (2) 

In fact, we shall define a linear system as one whose energy expressions 
are as given above. 

In addition to those systems that are genuinely linear many others can 
be considered approximately linear if their motions involve only small 
departures from their equilibrium configurations. To demonstrate this 
fact let us assume in what follows that the configurations of systems are 
determined by the generalized coordinates alone without having t entering 
explicitly. The potential energy of a system, being an energy of con- 
figuration, is then a function of the generalized coordinates, the q's. 

V = V(q v q 2 , ■ • • , q n ) 

Without losing generality, we may let q x = 0, q 2 = 0, . . . q n — 0, and 
V(0, 0, ... 0) = 0, when the system is at its equilibrium configuration. 
Thus the potential energy for any other set of values of the q's is, according 
to the mean value theorem, 

n [dV\ 1 n n I d 2 V \ 

Mi ( 3 ) 

in which the subscript of the first derivatives indicates that they are 
taken at the equilibrium position (0, 0, 0, . . . , 0) and the subscript of 
the second derivatives indicates that they are taken at some mean position 
(Wu W2» • • • ' *7 A)> w ^ tn tne ^' s lying between zero and unity. 1 

According to the principle of virtual work, when a system is in stable 
equilibrium its potential energy is at a minimum. Hence the first derivatives 
in (3) are zero. If we assume that the oscillation is small and the ^'s vary 

1 We assume that V is twice differentiable with respect to all the </'s and the second 
derivatives are continuous in the neighborhood of the equilibrium configuration. More 
often one considers (3) as the Taylor series expansion, neglecting terms of third and 
higher orders. In that case, the second partial derivatives will also be taken at </, = 
and Fwill have to be continually differentiable. 


in small ranges, the second derivatives can be considered as constants. 2 
Then V may be written 

1 n n 

v (<lv <l2>'- ,q n )=-22 CMi (4) 

^ i j 



c H (i,j= 1,2, ••*,») (5) 

This double summation in (4) when expanded is the same as (2). 

For the kinetic energy expression let us consider a generic mass element 
dm whose Cartesian coordinates x, y, and z are determined by the general- 
ized coordinates as follows : 

x = x(q v q 2 , • • • , q n ) 

y = y(qi, q*--, q n ) (6) 

z = z (qv q* • > q n ) 

Its velocity in the ^-direction, x, is then 

n dx 

* = Ip-4i ( 7 ) 

i dq^ 

o ™n dxdx 

x 2 = lZjr-2-qiqi (8) 

Similar expressions for y 2 and z 2 yield the kinetic energy of the system. 

T = - (x 2 + y 2 + z 2 ) dm 
2 Jm 

1 " * [ (dx dx dy dy dz dz \ i 

2 » j hi\dq i dq j dq i dq j dq^q^ 

Again, for small oscillations, the partial derivatives can be considered 
as constants with respect to / or the ^'s; and so is the last integral, being a 
definite integral over a fixed amount of mass M. 


T^lla.Mi (10) 

2 Perhaps we are putting the cart before the horse, and should have defined the word 
"small" to mean small enough so that the second partial derivatives may be considered 
as constants. 



f (dx dx dy dy dz dz \ 

Again, this is a repetition of (1). 

3.3 Summation Convention and Matrix Notation 

Two conventions will be adopted for use in carrying out our analysis. 

(a) the summation convention 

We notice, in all the equations studied so far, that whenever a certain 
subscript index is to be summed over all the available number of coordi- 
nates from 1 to n this index always appears twice in a given term. If this 
is always the case, the summation sign conveys no additional information 
and is therefore redundant. This is indeed generally the case for summation 
processes arising out of the types of operations we deal with. Thus the 
convention is hereby adopted that whenever a particular subscript index 
repeats itself in a given term a summation process is understood, and no 
summation sign will be used. Such an index is called a dummy index. In the 
meantime, a nonrepeating index is called a free index. It can take on any 
of the possible values 1, 2, . . . , n, but only one at a time. This convention 
is intimately connected with tensor calculus. The resulting notation may be 
called the tensor notation, although we may take it merely as a shorthand 
notation. For example, in the equation 

ft - c a a i 

i is nonrepeating, hence, a free index, whereas j is a dummy index. This 
equation is conventionally written 


fi = 2 c «fc ( /= 1,2, •••,/!) 


which represents a set of simultaneous equations written in "longhand 1 ' as 
/i = c n<7i + c i2<7 2 + ' * ' + c ln q n 

U = C 21?l + C 22?2 + * * ' + C 2n q n 

fn = Cntfl + Cn2 a 2 + ' * ' + C nn q„ 

When an equation is written with the convention described, a free 
index must appear precisely once in every nonzero term of the equation. 
It is evident that the indexing denotes only a certain pairing or grouping 


relationship. Whether an index is named i or / or others is of no conse- 
quence as long as the proper pairing relationship is kept. For instance, 
the following two equations are identical in every respect because they 
are identical when written in longhand. 

«< = c u<lj + bikPk 

a k = c k m q m + KiPi 

(b) matrix notation 

Only three kinds of matrices are needed in our analysis. The notations 
adopted are as follows : 

(i) An n x n square matrix is denoted by a bold sans serif letter in the 
upper case. The same letter in lower-case italic, with two subscript indices, 
denotes an element in the matrix. For example, A represents a square 
matrix, a 12 represents the element of A belonging to the first row and 
second column, and a {j is a generic element of A. 

(ii) An n x 1 column matrix is denoted by bold sans serif letter in 
the lower case. The same letter in lower-case italic, "with a single subscript 
index, represents an element of the column matrix. Thus a is a column 
matrix, a 2 is the second element of the column matrix a. and a { is a generic 
element of the column matrix a. 

(iii) A bar above a bold sans serif letter denotes the transpose of the 
matrix represented by this letter. Thus A is the transpose of A. and a 
is a row matrix which is the transpose of the column matrix a. A row 
matrix is always written as the transpose of a column matrix. 

(iv) Greek letters with or without subscript and italic letters without 
subscript denote scalar quantities. 

(v) In this type of analysis, the number n, which specifies the order of a 
matrix, is the same for all matrices within one problem. It is therefore not 
necessary to be specific. 

With the notation adopted, an equation written in matrix notation can 
be readily translated into one in the tensor notation. 

For example: 

Matrix Notation Tensor Notation 

a + b = ai + bi = 

A B = C a u b jk = c ik 

A i_ + g = aj] + gi = 

P A q = A p i a ii q j = A 

A B f + af = g "tfAi/ii + trfi = gi 

Note that in writing the tensor equation corresponding to a given matrix 
equation the last index of a matrix element must be the same as the first 


index of the following one. Similarly, to write the matrix equation from 
its corresponding tensor equation, the dummy indices should be arranged 
first to appear together. For example, 

a thifi = d k 
is first rearranged to give 

hfiufi = d k 
The corresponding matrix equation is 

BAf =d 
The same equation can also be written as 

fiflifiik = d k 

f AB = d 

This ambiguity is not important because the two matrix equations represent 
the same set of equations when written in longhand. 

3.4 Free Vibrations of an Undamped System — 
An Eigenvalue Problem 


In this article we shall carry out our analysis both in the tensor notation 
(on the left) and in the matrix notation (on the right) and repeat some of 
the more important results in longhand form. 

The energy expressions in (4) and (10) can be written with the tensor 
and matrix notations as 

T = iq A q 

-~ (11) 


y = \c ij q i q j 

The matrices A and C are called the inertia matrix and the elastic 
matrix, respectively. They describe the inertial property and the elastic 
property of the system. 

In free vibrations the principle of energy conservation demands that 

t + V= 

t = KMA + <h&Ah I r = Kq A q + q A q) 

Because both /and j are dummy Because t is scalar and A is 

and a u = a H , symmetrical, the two terms in 

the parentheses are equal. 

t = a ij q i q i t = q A q (A = A) 



V=c ij q i q j | K=qCq (C = C) 

The energy principle becomes 

(arf, + CyqMt = | q(A q + Cq) = (12) 

Since q t ; = or q = are trivial cases, we have 

a itti + c {j qj = | A q + C q = (13) 

In either form (13) represents the set of differential equations of motion 
conventionally written: 

Mi + c n?i + «i2& + Citfz + ' • ' + a ln q n + c ln q n = 

a 2±qi + Cffltfl + «22?2 + C 22#2 + " ' * + tf 2n ?„ + C 2n ?„ = 

a mqi + c nl ^ + a n2 q 2 + c n2 ^ 2 + • • • + a nn q n + c nB ^r fl = 

The a's are called the inertia constants and the c's, the elastic or stiffness 
constants of the system. 
To solve this set of equations, assume as usual a typical solution: 

q t = he i(ot | q = \e ia * (14) 

and substitute (14) in (13) 

(-w 2 ^,. + cM = | (-co 2 A + C) I = (15) 

This is a set of linear homogeneous equations in /_,. The case that l t = 
or I = represents the trivial case of no motion. However, these 
equations can have nontrivial solutions; that is, if the determinant of the 
coefficients vanish. 

| c .. _ oj 2 ^-| =0 | |C - co 2 A| = (16) 

Upon expanding the determinant, we obtain an algebraic equation in co 2 
of the nth degree, which is called the frequency equation or the characteristic 
equation of the system. It can be shown by theorems in algebra that since 
T and V are nonnegative the frequency equation will have n nonnegative 
roots. 3 Let these roots be 

co 2 = co ± 2 co 2 2 , • ■ • , co n 2 

They are the natural frequencies of the system. Let us assume for the 
time being the commoner situation in which these roots are positive and 
distinct. For each of the roots of co 2 it is possible to solve (15), in the 
sense that it is possible to find the ratio among the /'s. 

' 1 • '2 * ' 3 " * ' * " 'n = r l '- r 2 ' r 3 ' ' ' ' r n 

3 See Whittaker, Analytical Dynamics, Cambridge University Press, 1937, p. 183. 



h = Wi I I = fir 
where fx is an arbitrary complex number, and the r's form a column 
matrix whose elements are real numbers determined to within an arbitrary 
multiplication factor. The elements of r are real because the coefficients 
of (15) are real. Therefore 

q { = rtfie^ | q = rfie™ 1 

is a solution of (13), if (o satisfies (16) and r satisfies (15). The column 
matrix r specifies the amplitude ratio of a natural mode of vibration. 
It can be geometrically interpreted as a vector in fl-space whose components 
are the elements of the matrix r. Hence r is also called a modal vector. 

Since fi is arbitrary, it is not necessary nor convenient to have r include 
an arbitrary multiplication factor also. To remove this unnecessary 
ambiguity in r let us adopt a so-called normalizing procedure, whereby 
the first non-zero element in r is taken to be unity. In this way all the 
other elements of r are uniquely 4 determined. From here on we shall 
assume that the modal vectors are normalized. 

The basic procedure for determining the natural modes of vibration 
can thus be summarized as follows : 

(i) Choose a set of appropriate generalized coordinates and obtain the 
expressions for T and V, Lorn which the inertia matrix A and elastic 
matrix C are determined. 

(ii) Set up the frequency equation (16), using the inertia and elastic 
constants of the system determined above, and find the roots of the 

(iii) Put each root of the frequency equation into (15) and solve for 
/ 2 , / 3 , . . . , /„ with I ± = 1. (If the solution is infinite with / x = 1, set / x = 
and / 2 =1 ; then solve for the rest of the /'s. If this still does not give 
finite answers, set the next element of / equal to 1 , and so forth.) A solution 
obtained this way is the normalized modal vector r. 


Let the modal vectors corresponding to the natural frequencies of a 
system, oj v co 2 , . . . , co n , be represented by the symbols r (1) , r (2) , . . . , r (n) , 
respectively. The general solution of (13) is then 

q = pCi)^"* 1 + r^fi 2 e iu> ^ + • • • r^fi n e im ^ (17) 

4 We are assuming that the frequency equation has distinct roots. The case in which 
it has repeated roots is discussed later. Also, the normalization procedure described is 
one of the many possible. There is nothing physically unique about this procedure. 
See also Art. 3.12(b). 




where fA l9 ju 2 , . . . , p> n are arbitrary complex numbers which have to be 
determined by the initial conditions of the system. 

The expressions for this general solution can be condensed and brought 
into more familiar matrix and tensor forms by introducing the following 
two artifices. Let us first form a square matrix R, called the modal matrix, 
by putting together all the column matrices r so that theyth column in 
R has the same elements as r {j) . Or 

' 10 ' I 


[ r <u 



Then let us form a column matrix p whose elements are given by 

P = 

The solution can then be written simply as 

Q i = r aPi I q = R p 
?i(0 = >n/>i(0 + 'lW) + " " " + r ln p n (t) 
ft(0 = *Wi(0 + r^pzit) + ■ • • + r 2n p n (t) 


>1^ V1 



ju 2 e iaj ^ 






<{JS) = r mPi(t) + r n2 p 2 (i) + ■ • • + r nn p n (i) 
The set of functions p { (t) is defined either by (19) with complex number 
representation or more conventionally by 




I//J cos (o^t — 04)" 
\fi 2 \ cos (o) 2 t — a 2 ) 

\f*n\ cos (<o n t - a„) 


The relationship between ju, \ju\, and a is 

To obtain the particular solution satisfying a given set of initial condi- 
tions, these constants can be determined by the following procedure. 
Let q(0) and q(0) be given. Put these into (20) to solve for p(0) and 
and p(0). 

qm = r ijP m 

qM = r ti p0) 

q(0) = R p(0) 
q(0) = R p(0) 


where p{t) is defined by differentiating (19) or (21). When p(0) and p(0) 
are found, the constants ju, \/u\, and a can be determined from (19) or 



The method described for finding the frequencies and modal vectors 
becomes impractical when the number of degrees of freedom n is large. 
The number of arithmetic operations needed increases very rapidly with n. 
To devise other methods, it is necessary to introduce a new concept, 
the concept of an eigenvalue problem. 

Stated in its most general form, an eigenvalue problem is a mathematical 
problem containing an unspecified parameter in such a way that the 
problem has only a trivial solution, unless this parameter takes on one 
of a certain set of values called the eigenvalues or characteristic values of 
the problem. The simplest and most important type of eigenvalue problem 
is illustrated by the following example. Given a square matrix L, which 
represents a linear transformation of vectors in fl-space, we are to find a 
vector r that is transformed by L into another vector in the same direction. 
In other words, the transformation L changes only the length but not the 
direction of r. Such a vector is called an eigenvector of L. The condition 
to be satisfied is represented by the equation 

Lr = Xr 

The scalar quantity X represents the ratio between the lengths of the vectors 
before and after transformation, and it has to take on one of the eigen- 
values for eigenvectors to be found. It is evident that eigenvectors are 
determined only to within an arbitrary multiplication constant. However, 
they can be normalized as desired. 

The problem of determining the frequencies and amplitude ratios of the 
natural modes of vibration can be stated as an eigenvalue problem. 
Evidently the modal vectors satisfy (15) 

(-w 2 A + C) r = 
This can be written 

(A- 1 C) r = co 2 r (23) 

Hence the modal vectors and the natural frequencies are the eigenvectors 
and eigenvalues of a linear transformation represented by the matrix 
product (A- 1 C). Or, L = A 1 C and X = co 2 . 

The problem of finding the eigenvalues and their associated eigenvectors 
is an integral problem. It is not necessary for one to be found before the 
other. In the classical procedure the eigenvalues are found first. There are 
other procedures in which the reverse is true or the eigenvalues and eigen- 
vectors are found simultaneously. 

A matter of practical interest is the fact that finding the eigenvalues and 
eigenvectors of a matrix has become a routine problem for electronic 


digital computers. In most computing laboratories the programming for 
solving such a problem is readily available. 

3.5 Principal Coordinates and Orthogonal 
Property of Modal Vectors 

(a) a transformation into principal coordinates 

We have stated in Art. 3.1 that any set of quantities can be used as a 
set of generalized coordinates of a system if by specifying their values the 
configuration of the system is determined. It follows that ifq l9 q 2 , . . . , q n 
are a set of generalized coordinates of a system any other set of quantities 
that will determine the ^'s uniquely can also be used as a set of generalized 
coordinates. According to (20) the set of quantities p ± , /? 2 , . . . , p n deter- 
mines q ± , q 2 , . . . , q n ; hence the /?'s can also be considered as generalized 
coordinates of the system. Obviously, when the coordinates used to 
describe a system are changed, the differential equations will be changed 
also. Now we want to see how the equations change when the coordinates 
are changed from q v q 2 , . . . , q n to p l9 p 2 , . . . ,p n . The matrix q satisfies 

Aq + Cq = (13) 

The matrix p then must satisfy 


This equation may be premultiplied by (AR) _1 to yield 

p + (A R)-i C R p = 

p + R -i A 1 C R p = 

Let us define a matrix W 2 as 

W 2 = R^A^CR 

Then p satisfies 

p + W 2 p = (24) 

In the meantime it is known that if q satisfies (13) and p is related to q 
by (20) p l9 p 2 , . . . , p n are a set of sinusoidal functions satisfying 

Pi + M i 2 Pi = ° 
p 2 + co 2 2 p 2 = 

Pn + M nPn = 


This set of equations can be written in matrix form as 






.Pnl L ° ° '"CO, 

Comparing this equation with (24) we have ; 







R- X A -iC R = W 2 




In other words, W 2 is a diagonal matrix, and the transformation of the 
coordinates from the q's into/?'s results in a separation of the variables in 
the differential equations of motion. The /?'s are therefore called the 
principal coordinates. 6 

It must be borne in mind that the relationship (25) is one between a 
matrix A -1 C, its eigenvectors which form the matrix R, and its eigenvalues 
which form the matrix W 2 . It is a mathematical relationship not neces- 
sarily connected with the phenomenon of free vibration. For instance, in 
a forced vibration problem we shall see that the matrix equation is the 
nonhomogeneous equation 

A q + C q = f (26) 

The transformation procedure that leads from (13) to (24) transforms this 
equation into 

p + W 2 p=R 1 A- 1 f (27) 

in which W 2 is still a diagonal matrix and the unknowns p^p 2 , • • • ,p n 
are separated in the set of differential equations represented by the fore- 
going matrix equation. 

(b) orthogonality of modal vectors 

A somewhat more convenient procedure of transformation into the 
principal coordinates is based on the so-called orthogonality properties 

5 We also have to take cognizance of the fact that them's are linearly independent in 
order to arrive at (25). 

6 If a certain normalizing procedure, such as the one described in the Art. 3.4, is used, 
the principal coordinates become the normal coordinates. 


of modal vectors. Two vectors a and b are said to be orthogonal with 
respect to a symmetrical matrix S if the product 

a Sb = b Sa = 

Now we want to show that any two modal vectors describing two natural 
vibrations of different frequencies are orthogonal with respect to both 
the inertia matrix A and the elastic matrix C. Let r U) and r (i) be two 
modal vectors. They satisfy 

cofA r«> = C r« 

co/A r<>> = C r<« 

co, =£ to, (28) 

Premultiply the two equations by r 0) and r (0 , respectively. 

wXr™ A r<*>) = r^ C r™ 

cofir™ A r (y >) = F (i) C r< J > 

Because both A and C are symmetrical, 

pW) C r (() = r (? '> C r (;) 
pw) a r< ? > = F< ? > A r<« 

Upon subtracting the two equations in (29) from each other, we have 

(w, 2 - cofiCf^A r<*>) = 

Since oj t ^ co^ we have the proof that 
fii) A r «) _ 

?wcpW-0 "'*"' (31) 

In tensor notation these two equations are written 

r k a km r m — u , 

f\ t-\ c °i ^ c °i 

r k C km r m — U 

In either way of writing the equations contain only scalar products 
involving modal vectors. The same orthogonality property can also be 
described in terms of the modal matrix R given by (18). Let us form the 
matrix products 

r k PkrJmi = ™,i R A R = M 

r kfkm r mi — kji R C R = K 

We can see from (18) and (31) that if the natural frequencies for different 
modes are. distinct the matrices M and K will have only diagonal terms. 
That is, 

m {j = k tj = when / =£y* 


The transformation of coordinates from q's to /?'s can now be performed 
on (13). Substitute (20) into (13) and premultiply the equation by R. 
The result is 




M p + K p = (32) 

Since M and K have only diagonal terms, the longhand expression for 
(32) is 

™iipi + k nPi = 

m 22p2 + k 22 P2 = 

m nn p n + k„ n p„ = 
Evidently, since p is given by (19), 

^n 2 22 2 7 

m u «7 2 2 ;77 7 

This can also be arrived at by the following operations: 

M 1 K = (R A R)-i(R C R) = R 1 A 1 R - 1 R C R 

= R-i A^C R = W 2 (33) 

The transformation from (13) to (32) is called orthogonalization. 

The mathematical problem of determining the natural frequencies and 
modes of vibration is also one of finding a matrix R which will diagonalize 
two given symmetrical matrices A and C through 

RAR=M and RCR=K (34) 

and will give the natural frequencies through 

M 1 K = W 2 (35) 

This approach is used in some numerical methods of solving the problem 
by successive approximation. 

3.6 Rayleigh's Quotient 

In Art. 3.5 the two equations (34) and (35) imply that if r is a modal 
vector the quotient 


is the natural frequency of vibration associated with that modal vector. 
Let us now investigate the nature of such a quotient formed with an 
arbitrary vector u. This quotient is called Rayleigh's quotient of u. 

u C u 

fi(u) = =—- (36) 

u A u 

Let us retain the assumption that the frequency equation has distinct 
roots. With this assumption, it is not difficult to show through the 
orthogonality property of the modal vectors that these vectors are linearly 
independent. (See Exercise 3.10.) An arbitrary vector u in «-space can 
thus be expressed in terms of a linear combination of the modal vectors 

u = pjpff) + v 2 r {2) + * * ' + v n r {n) 

in which v l9 v 2 . . . , and v n are a set of coefficients. By utilizing definition 
(18), we can rewrite this equation in a form similar to (20): 

Ui = r ijVj | u = R v (37) 

The meaning of (37) is, however, slightly different from (20), since (37) 
does not contain the time variable /. 

Rayleigh's quotient formed with the vector u can thus be expressed 

vRC Rv VKv 

eoo = - BAP = =r— (38) 

vRARv vMv 

Since K and M are diagonal matrices and K = M W 2 , we may write (38) 
in longhand form : 

n - m uV ( °i 2 + ^22^2 2 ^2 2 + • • • + m nn v n 2 co n 2 

t^ 2 i 2 i i 2 W"/ 

m^vf + m 22 v 2 2 H + m nn v n z 

Consider Q as a function of n variables, i\, v 2 , . . . , v n , defined by 
(39). Since the denominator in (39) does not vanish except when i\ = 
v 2 = . . . = v n = 0, the function is continuously differentiable except at 
the origin. Hence the function possesses a Taylor series expansion, and 
for small changes in the y's the change in Q is given by 

AS = IP &v t + I J!f- A,, A., + • • • (40) 

in which the summation convention is employed. 

Let us assume now that u coincides in direction with one of the modal 
vectors, say r (1) ; then u = v x r a) and v 2 = 0, i? 3 = 0, . . . , v n = 0. Through 
routine differentiation of (39), it can be shown easily that under these 

££ = |e = 0,..,§e = (41) 

ov 1 dv 2 dv n 


Similarly, these partial derivatives vanish when u coincides with any 
modal vector r (i) . Hence Rayleigh's quotient achieves a stationary value 1 
at any of the modal vectors. Furthermore, if 6o x < co 2 < ... < co ni w^ 
is the absolute minimum of Q. In other words, Rayleigh's quotient is 
always larger than or at least equal to the lowest natural frequency of the 
system. (See Exercise 3.11.) 

By substituting (41) into (40), we obtain the relation 

2 ov t ovj 

which holds in the neighborhood of a modal vector. This relation indicates 
that if the difference between a vector and a modal vector is small, small 
changes in this vector will produce only second-order changes in the value 
of the corresponding Rayleigh's quotient. Since the value of Rayleigh's 
quotient for a modal vector is the square of the natural frequency of the 
corresponding mode, a vector that is a first approximation of a modal 
vector will yield, through its Rayleigh's quotient, a better approximation 
of the corresponding natural frequency. This approximate method of 
finding the natural frequency is called Rayleigh's method. 

3.7 Forced Vibration of an Undamped System 


Since the coordinate system used in this analysis is in a generalized sense, 
it becomes necessary to have the corresponding concept of generalized 
forces. Let us first be reminded that our analysis is based on the study 
of energy forms, and our equations are derived from energy expressions. 
It is therefore obvious that those external forces which act on the system 
but do no work do not enter into the picture. Forces of this type are called 
forces of (nondissipative) constraint. Forces whose work upon the system 
is included in the potential energy expression are called potential forces and 
need not be considered again. 8 Therefore, only the forces that do work 
but are not accounted for in the potential energy of the system need to be 
added to the equations of motion (13). 

Since the consideration here is energy, it is natural to define generalized 
forces by the work they perform on a system. The power, or the time rate 

7 A stationary value can be a maximum, a minimum, a minimax, an hyperbolic point, 
or others. 

8 For example, the gravity force does work on a simple pendulum; but this work is 
the potential energy of the system, and the force is taken into account in the change 
in potential energy. 


of doing work, by external forces is a linear function of the velocity 
components of the mass particles on which the forces are acting. According 
to (7) the velocity components themselves are linear functions of the qs. 
Hence the power of external forces may be written as linear functions of 
the q's: 

Mi = Mi + Mi + • • • + Mn (42) 

The coefficients /'s are called the generalized forces with respect to the 
coordinate system q t . For linear systems and for systems in small oscil- 
lations these coefficients can be considered as independent of the q's and 
are functions of time t only. 

With generalized forces acting on the system, the energy equation 

t+V=f i q i (43) 

By following the same steps in the derivation of (13), we arrive at 

<*<A* + c iAi =fi I A q + C q = f (44) 

(b) steady-state response to harmonic and periodic 
generalized forces 

The special case in which every f t is a harmonic function of time having 
the same frequency co f is now considered. 
Let the generalized forces be represented by 

ft = gie""'' | t = pf* 

In this case the solution consists of a steady-state vibration of frequency 
(o f and transient vibrations with the natural frequencies of the system. 
We are interested only in the former. Let the steady-state vibration be 

q. = //<V | q = \ e i°>ft 

By putting this into (44) and canceling out e lC0ft , we have 

(- m t *a„ + c«)/,= gi | (-<A+C)l = g (45) 

This is a set of linear simultaneous equations that can be solved by standard 
methods. If m f is equal to any one of the natural frequencies, resonance 
takes place. If f is a periodic force, it can be expanded into a Fourier 
series, and the response to f can be obtained by the method of superposition 
similar to that described in Art. 1.8. 


For the general solution of a forced vibration problem let us begin with 

Aq + Cq = f 


Transforming it into the principal coordinates, we have 

Premultiply both sides by R and utilize (34) to obtain 

Mp + Kp = Rf 



p + W 2 p = M -i R f 



With M a diagonal matrix, M -1 is easy to compute since it is also a 
diagonal matrix with elements m^ 1 = l/ra n , m 22 = l/^ 2 2> . . . , m~^ — 
\jm nn . It is also easy to show that (47) is the same as (27) by deducing 
from (34) that 

M-!R = (A R) 1 

Returning now to (46) and writing it in longhand form, we have 

™nPi + KiPi = hifi + r 21 f 2 + * ' ' + r nl f n = j\(t) 
m 22 p 2 + k 22 p 2 = r 12 f t + r 22 f 2 + h r n2 f n = j 2 (t) 

m nnPn H" ^nnPn = r lnfl + r 2nj2 + * ' + f n njn = JnV) 

These equations can be solved separately. For instance, according to (76) 
in Art. 1.10, if the system is initially quiescent, the solution for p x is 


Vm u k n J 

j\(t) sin co^t — t) dr 


AW = 'li/iO) + r 21 f 2 (r) + • * ' + r nl f n (r) 
Let us define a matrix 

sin W/ = W/ - -i W 3 / 3 + ~ W 5 r 5 • • • 



Since W is a diagonal matrix, the series above also represents a diagonal 
matrix, which is 

sin co x t ••• 

sin Y/t = 

sin co 2 t 

sin co J 


The solution to (46) or (47) can thus be written 

p(f) = J (M K)"* sin W(f - r)j(r) rfr 


= (M K)"* J sin W(f - t)R f(r) rfr (51) 

if p(0) = 0andp(0) = 0. 

The solution for q is therefore 

q(0 = R p (0 = R(M K)-y* I sin W(f - t)R f(r) Jr (52) 


if q(0) = and q(0) = 0. 

On the other hand, if the initial values do not vanish, the following 

terms must be added to (51) and (52), respectively: 

cos V/t p(0) + W- 1 sin V/t p(0) 

R cos V/t R- 1 q(0) + R W 1 sin Wr R- 1 q(0) 

The reader is left with the proof of this statement. (See Exercise 3.17.) 

3.8 Free and Forced Vibrations of a Damped System 

(a) free vibrations 

When there are viscous forces acting on certain parts of the system, 
there is an energy dissipation. It is not difficult to see that if the forces 
are proportional to the velocities or relative velocities at certain parts of 
the system the rate of local energy dissipation is proportional to the 
square of the velocity of the parts affected by the damping force. This is 
true whether the damping forces are external or internal. According to 
(7) and (8), the squares of the velocities are quadratic functions of the 
generalized velocities q's. Hence the rate of energy dissipation for the 
entire system can be written as 

2F=b ii q i q i | 2F = q B q (53) 

The energy equation for the system is then 9 

f + V + 2F = 

Following the same argument used before, we arrive at 

*<Ai + Mi + Wi = ° I Aq + Bq + Cq = (55) 

9 Fis a scalar function called Rayleigh" s function. The factor 2 is included to give F 
the same appearance as T and V and also to give certain symmetry to Lagrange's 
equation, modified by Lord Rayleigh. 


Let the solution be of the type 

ft = he"' | q = \e°> 

We have 

(j"fl w + sb {j + c w )/, = | (s 2 A + sB + C)l = (56) 

For nontrivial solutions of (56) the determinant 

D(s) = \s 2 aij + sb ti + c w | = (57) 

This is an algebraic equation in s of 2«th degree. It has In roots, which 
may be real or complex. The rest of the procedure is similar to that 
discussed in Art. 2.7. The results can be stated as follows. 

To each solution of (57), say s = s {i) , there is a vector \ (i) such that 

q = |<V^ 

is a solution of (55). There are In such solutions, and the general solution 
of (55) is therefore 

q = f I'V^ (58) 

If s {i) is real, \ {i) has real elements. If it is complex, it must come as one 
of a complex conjugate pair, and the vector I's associated with this pair 
also form a complex conjugate pair. The real I's are determined only to 
within an arbitrary real multiplication factor, and the complex I's are 
determined only to within an arbitrary complex multiplication factor. 
In any event, there are In integration constants to be determined by the 
In initial conditions. 


Forced vibrations of damped systems are described by a set of equations 
having the general form 

or (59) 

A q + B q + C q = f(r) 

(i) Steady-state response to harmonic forces and transfer functions. If 
f(t) is a sinusoidal function of time 

fi = g/ Mft I f = & iWft 

the steady-state solution of (59) can be obtained by assuming that it has 
the form 

q. = l/^ | q = \e iu >f l (60) 


This results in a set of linear simultaneous equations represented by 

(_ W/ 2 A + /ft)fB + C )l = g (61) 

Let us define a matrix function 

D(ioj f ) = -co f 2 A + ico f B + C (62) 

The solution to (61) is then 

I = D-\ko f )g (63) 

provided that the determinant 


Otherwise steady-state solution is not possible, for the amplitude tends to 
grow without limit, and resonance is said to exist. 

A problem of particular interest is that in which all but one of the g's 
vanish; that is 

gi = g% = ' ' ' = gi-l = gj+l = * * = gn = 

gi = 1 (64) 

In other words, g is equal to the unit vector of they'th coordinate. Let us 
call the amplitude of the response of a generic coordinate q t to this unit 
harmonic force the transfer function t i} . Evidently, the different transfer 
functions between the different pairs of coordinates can form a matrix. 
The elements of this matrix are obtained by substituting (64) into (63), 
and it is not difficult to see that this matrix is the inverse of the matrix D. 

To extend the domain of definition of the functions into the complex plane, 
we replace the argument ico f by the complex variable s. Hence 

T(s) = D~\s) = (5 2 A + *B + C)- 1 (65) 

This transfer function matrix is useful in studying the transient response 
of a damped system. 

If the forcing functions represented by f are periodic, we can expand 
them into Fourier series and then apply the principle of superposition to 
obtain the steady-state solution in the usual way. 

(ii) Transient response — indicial response and impulse response. The 
transient response of a damped system can be investigated through its 
indicial response and impulse response, as discussed in Art. 1.11. Similar 
to the situation with the transfer functions, for systems with many degrees 
of freedom there are many indicial response functions that can be defined. 
Let us use the symbol i$\t) to denote the indicial response of a generic 


coordinate q i to a unit step force applied to theyth coordinate. In other 

ft = «P(0 I q = "<'>(*) 
satisfies (59) under the conditions 

/i-/»-"- =/*-i =/m = •••=/„ = 
and (66) 

fi = J"(0 

together with the initial conditions 

?,(0) = 
WO) = o 

q(0) = 
q(0) = 


When the indicial response functions are continuous for t > and 
differentiable for / > 0, the associated impulse response function can be 
obtained by differentiation with respect to t. 


h< J ')(0 = T u(i) (0 

According to the discussions in Art. 1.11, the general solution to (59) 
under the initial conditions (67) can now be constructed from the impulse 
response functions: 

q(0 = |W)h H, (/ - t) +/ 2 (r)h'»(« - t) ■ ■ -/„(r)h<">( ( - r)] dr 


JO j = l 

To simplify this expression, let us form two square matrices U and H 
whose columns are the vectors u's and h's, respectively; that is 

»« = ^ h ti = up 


The solution to (59) can then be written 

qlt) = fhtfi - t)//t) rfr 

« ^> 

q(0= H(/-r)f(r)^T 


The impulse response matrix H can be obtained by differentiating U, 
the indicial response matrix, which is formed by the solutions to (59) under 




the conditions (66) and (67). The matrix H can also be obtained by 
integrating the transfer-function matrix T(s) with the integration factor 
e st , as described in Art. 1.11(d). 

d 1 f c+ix 

H(0 = -U(0 = — T(s)e«ds (69) 

at Itti J c -i oo 

in which c is any positive constant. 

It is to be noted that in analyses of the vibrations of damped systems we 
do not try to orthogonalize the equations by a suitable linear transfor- 
mation of the coordinates, as we do with undamped systems. There is an 
orthogonalization procedure, 10 but it is seldom utilized because of compu- 
tational difficulties. 

3.9 Semidefinite Systems 

Most vibratory systems are connected with a stationary frame; as a 
matter of fact this stationary frame may be considered a part of the system. 
For such systems the equilibrium configuration is stable, and any variation 
in the generalized coordinates must be accompanied by a change in the 
potential energy of the systems. These systems are called positive-definite 
because their potential energy expression 

V = i q C q > (70) 

except at the equilibrium configuration 

q = 

On the other hand, a system is called semidefinite if it is possible for V 
to vanish without having all the q\ vanish simultaneously. la other words, 
q = corresponds to a condition of neutral equilibrium. 

Mathematically, the necessary and sufficient condition for the potential 
energy expression to be nonnegative is that 

c w c \i c \z 

•ii >o 



c 12 
C 22 


C 12 

C 2 2 
C 23 

C 23 
C 33 


ICl = 


10 See "Coordinates Which Uncouple the Equations of Motion of Damped Linear 
Dynamic Systems" by K. A. Foss, Journal of Applied Mechanics, Vol. 25 (1958), 
pp. 361-364. 




For positive-definite systems all of the determinants are greater than zero 
and for semidefinite systems the last one or more of the determinants are 
equal to zero. In vibration study it is seldom necessary to evaluate these 
determinants, since an examination of the make-up of the system is usually 
enough to determine whether or not it is semidefinite. If the system does 
not include a stationary frame, or the stationary frame is not positively 
connected to some part of the system, the system is semidefinite. 11 Two 
such systems are illustrated in Fig. 70. 










Note that we are not concerned about the kinetic energy expression, 
since any motion of a system must result in a positive amount of kinetic 

The analysis presented so far in this chapter is essentially valid for 
semidefinite as well as positive-definite systems, but there are certain 
features of semidefinite systems that we shall discuss. 

Since for a semidefinite system 

M = o 

one or more vectors I exist, so that 
c»h = 

IC! = 




This means that (15) can be satisfied with to = 0, and (13) can be satisfied 


/,(** + p) 

q = Kaf + p) 


11 We assume that the equilibrium configuration is at least not unstable so that the 
potential energy is nonnegative. 


which represents motions having constant velocities. Semidefinite systems 
are thus associated with zero roots of the frequency equation (16). The 
number of independent nonzero I's which satisfies (72) is the same as the 
multiplicity of the root co 2 = in (16). Together, they represent some 
of the natural modes of the system, which may be called the zero modes. 

In mechanics there are two types of problems involving the study of 
semidefinite systems. In the first type we are given systems known to be 
semidefinite, systems such as a rotating shaft, a moving vehicle, or a 
rocket in free flight. We are to analyze their vibrational modes in addition 
to the known zero modes. In the second type of problem the potential 
energy expression V or the elastic matrix C includes some parameters 
that must be determined so that the system can be made semidefinite. 
This is the type of problem studied in theory of elastic stability. 

For a positive-definite system the elements in the inverse of its elastic 
matrix, C -1 , are known to be the influence coefficients. For a semidefinite 
system the matrix C is singular because of (71), so that its inverse is not 
defined. This agrees with the physical facts that when a system is in neutral 
equilibrium its influence coefficients are mostly undefined quantities. 
(See Exercise 3.15.) This has two practical consequences. The first is 
that we cannot write the alternate form of (23) 

[C-i A]r = 1 r 


which is a desirable form for some methods of numerical computation, 
discussed in Art. 3.11. The second is that without the use of influence 
coefficients it becomes inconvenient to set up the equations of motion for 
some systems, such as a beam carrying three masses in free flight. 

The zero modes of a semidefinite system are not of interest in vibration 
study. It is possible to "suppress" these modes by introducing constraints, 
which will reduce the number of degrees of freedom and in the meantime 
convert the system into a positive-definite one. This procedure is illus- 
trated in Art. 2.9(a) and may be generalized as follows. Let 1° represent 
a zero mode. Then 

Cl° = 
Since C is symmetrical, 

i°c = o 

Premultiply (13) by 1° to obtain 

T°Aq + !°Cq = 


1° A q = i°Aq = af + /> 




Suppressing the zero mode is equivalent to letting a = and b = 0. 
In other words, only those motions orthogonal to the zero mode are 
considered. This results in a constraint relation between the g's. Written 
longhand, the relation is 

('All + / 2 %j + • • • + />m)?l + (/xV + / 2 °«22 + ' * ' + '>2n)?2 

+ ' ' • + (h°a ln + l 2 «a 2n + ' ' • + l n «a nn )q n = (74) 

For each zero mode there is a constraint relation such as the one above. 
With these constraints, a corresponding number of the coordinate variables 

J -T 



Figure 71 

g's can be eliminated. This elimination process results in a positive- 
definite system having fewer degrees of freedom. 12 

It is not necessary that the constraint which suppresses the zero mode 
be obtained by finding nontrivial solutions of (72). In many cases it can 
be obtained by simple principles of mechanics. As an example, let us 
consider the free vibration of a light rectangular beam carrying three 
particle masses, as shown in Fig. 71. The beam is unsupported in free 
space. This vibration is often called the free free-vibration. Obviously, 
the degrees of freedom consist of three translations, three rotations, and 
two vibrations in the two principal planes of the beam cross section. Of 
the eight degrees of freedom, only two are nonzero modes. To solve this 
problem, we can suppress all the zero modes by assuming that the system 
has no linear momentum and no angular momentum. Furthermore, we 
restrict the motion in one of the two principal planes of bending of the 
beam. This is permissible, since obviously there is no coupling between 
the two vibrations in the two planes of bending. Referring to Fig. 71, we 
have therefore 

mtfi + m 2 q 2 + m 3 q 3 = 
m#! • (2a) + m 2 q 2 • (a) = 

For an illustration of a similar problem see Art. 3.11(e). 


Upon integration and setting the integration constants equal to zero, 
we have 


T= 2 ( m &* + m2 ^ 2 + m ^ 

1 / _ 4m] 2 

1 1 48 F/ 

K= 2 W2 = 2W [K?1+?3) - ?J! 

2 \ m<> mo/ 


.2 \ mo/ 777o . 

?! 2 

where i^ represents the bending stiffness of the beam cross section. For 
the vibration in the other plane of bending we need only change I ± to L 2 
in the potential energy expression. 

3.10 Repeated Roots of the Frequency Equation 

In Art. 3.5 it was shown that if the roots of the frequency equation 

|C - w 2 A| = 

are distinct there is a modal vector to each of these roots. These modal 
vectors are orthogonal to one another, with respect to both A and C. 

po) A r<« = 

CO; ^ (Oj 

pw) c r<*> = 

The orthogonal relationship among the modal vectors also insures that 
they are linearly independent. 

Now if some of the roots of the frequency equation repeat themselves, 
we will have fewer distinct natural frequencies. The question is then 
"Does the system have fewer natural modes?" The simple examples 
shown in Arts. 2.10 and 2.14(a) indicate that the answer is negative. 

A rigorous mathematical treatment of the problem is not necessary for 
our purpose. We simply state that in the eigenvalue problem (23) arising 


from free-vibration analysis the coalescence of eigenvalues does not cause 
a coalescence of eigenvectors. This statement, however, is not generally 
true for an arbitrary eigenvalue problem. 

Lr = 2r 
To summarize, let 

be an Mime repeated root of the frequency equation (16). There are r 
linearly independent modal vectors, r (1) , r (2) , • • • , r (r) , representing 
different modes of free vibrations having the same frequency. These 
modal vectors are orthogonal, with respect to A and C, to all other modal 
vectors of the system not belonging to this set. Among themselves the 
orthogonality relationship may or may not hold. However, the set of 
modal vectors corresponding to a given repeated root are not uniquely 
determined because any linear combination of these vectors can also 
serve as a modal vector, since 

(A- 1 CXajH" + a 2 r< 2 > + • • • + a r r (r) ) 

= co r 2 (air (1) + a 2 r< 2 > + • • • + a r r<'>) 

It now becomes possible to construct a new set of modal vectors by taking 
different linear combinations of the old set, so that the vectors in the new 
set are orthogonal among themselves with respect to A and C. The method 
of construction is left to the reader as an exercise. (See Exercise 3.16.) 


3.11 Solution of Eigenvalue Problems by Matrix Iteration 

The direct method of solving the eigenvalue problem 

(0 2 A r = C r 

as outlined in Art. 3.4, is not practical when a large number of coordinates 
is involved because of the excessive amount of arithmetic computations 
needed. There are several numerical procedures that simplify computa- 
tional steps in one way or another. Aside from the amount of labor 
involved, the question of checks, control of round-off errors, susceptibility 
to mistakes, etc., must also be considered. Complete solution of large- 
scale eigenvalue problems is practicable only through the use of electronic 
digital computers. Numerical procedures, which are suited for program- 
ming on such computers, have requirements of their own, but this is a 
subject in itself and cannot be dealt with exhaustively here. In this article 


we can discuss only a few methods that may be used to solve problems of 
reasonable size with the help of a desk calculator and slide rule. 

(a) determination of largest eigenvalue by iteration 

The basic principle of the matrix iteration procedure is extremely 

Let the eigenvalues of a matrix L be 

?i 1 , X 2 , • - • , ). n 

\K\ > N > \h\ ■ ■ ■ \K\ 

Designate the corresponding eigenvectors by 

r (D r (2) . . . r (n) 

Let v be an arbitrary vector. It can be thought of as a linear combination 
of the eigenvectors, since the eigenvectors are linearly independent. 

v = ai r (1) + a 2 r (2) + • • • + y. n r {n) 

Lv = a 1 L r<« + a 2 L r< 2 > + • • • + a B L r<»' 

= OjL^rW + a^,r«» + ■ • • + a n A„ r<»' 
By repeating this process, we have 

L 2 v=LLv = a 1 A 1 2 r<D + a 2 A 2 2 r< 2 > + • • • + a n A B 2 r<») 

L* v = a 1 A 1 V (1) + a 2 A 2 l > (2) + • ■ * + «„;„><*) 

W > 1^1 > • • • |4| 

as / becomes sufficiently large, the first term on the right-hand side 
predominates over the rest of the terms and 

Thus U v is parallel to the eigenvector r (1) , and the corresponding eigen- 
value X x can be obtained by operating it once more by the matrix L. 

We thus have a method of determining the largest (in absolute value) 
eigenvalue and the associated eigenvector of a matrix L. The method 
consists of repeated multiplication of an arbitrary vector by the matrix L. 
The rapidity with which the eigenvector r (1) is approached depends upon 
two things: the closer the starting vector v parallels r (1) , the faster the 
convergence. In other words, it is desirable that the coefficient x 1 be 




larger than the rest of the a's. Theoretically, there is a possibility that by 
accident a starting vector with a x = will be chosen. Then the iteration 
process described will converge to the eigenvector with the next largest 
eigenvalue. Practically, this possibility does not exist, for even if a x = 
in v, unavoidable round-off errors will introduce a small component 
along r (1) in the iterated vector, and this component will eventually 
dominate the rest. The second thing that enhances the rapidity of con- 
vergence is to have the eigenvalue X x much larger in absolute value than 
the rest. This is especially important in large matrices. 

Let us now illustrate the procedure by a simple example. One small 
modification will be made in the illustration. To keep the numerical value 
of the elements of the iterated vectors within a certain range, they are 
normalized after each iteration. A self-explanatory tabulation scheme is 
also used. 










































L v iv 

v v 

Lv v 































Thus the approximate answer is 

X x = (7.98 + 8.01 + 8.01)/3 = 8.00 
pa) = [1.00 1.98 -4.96] 
The exact answer of this problem happens to be 

l x = 8.00 
F< 1} = [1.00 2.00 -5.00] 

The procedure illustrated is straightforward. Such routine calculation 
can be put into the hands of untrained office help. On the other hand, if a 
little ingenuity is exercised, the process can be speeded up. For instance, 
we must realize that each iterated vector is a trial vector for the next step, 




so that it may be changed at will. After a few steps, a trend of changes in 
the elements is discernible. The trial vector of the succeeding step can be 
altered judiciously from that given by the previous step to hasten the 

The matrix L used in this example was obtained from the following 
problem : 

T = |(21<7i 2 + 19<? 2 2 + 5q z 2 + 14^ 2 + \4q 2 q 3 + 2q Y q z ) 
V = i(14^ 2 + 6? 2 2 + 2q 2 + 6 Ma + 2q 2 q 3 - 4 Ma ) x 10 3 

x 10 3 

A- 1 C= 125 x L 

The reader will profit by solving this eigenvalue problem by the classical 
method. The answer is 




r l4 






C = 










(0 ± 2 = 


r n 

r (i) = 




r<« = 



r^ 6 ' ^ 

(b) determination of the lowest natural frequency 

In the majority of vibration problems it is the lowest natural frequency 
that is of interest, whereas the method illustrated gives only the highest 
eigenvalue. This situation can easily be remedied if our eigenvalue problem 
is formulated as 

(C-i A) r = - r 

CO 1 

The largest eigenvalue of C _1 A will then correspond to the lowest natural 
frequency. In this formulation the system must be positive-definite so 
that C _1 is defined. 13 

Example: Given 

T = «3& + 4^ 2 2 + 2^ 3 2 + q, 2 + 4q,q 2 + 2^ 3 + 2q 3 q A ] X k t 
V = M4?i 2 + 5</ 2 2 + q 3 2 + 2? 4 2 - 4q x q 2 - 2q 2 q 3 ] X k v 

in which k t and k v are constants having appropriate units. 

13 For iteration of a semidefinite system see "Iteration in Semi-definite Eigenvalue 
Problems" by B. M. F. de Veubeke, Journal of Aeronautical Sciences, Vol. 22, No. 10, 
October 1955. Also, many physical problems are formulated by influence coefficients 
to begin with. The influence coefficients form the matrix C _1 and can be used directly 
in the ensuing computation. 


To find the lowest natural frequency and the associated modal vector 

A = 












x k t 










x k, 

C-! = 



































The iteration of the last matrix: 






























9 3 


12 6 


18 18 



21 1.0 

27 1.0 


27 1.3 

38 1.4 


51 2.5 

79 2.9 


6 0.3 

8 0.3 





















A note about another scheme that often simplifies computation. We 
observe that in the foregoing tabulation 


L a = b and L c 
L (c - a) = d - b 
d = b + L (c - a) 

Since (c — a) is small, d can be conveniently obtained by applying a smal 
correction to b. Similarly, the next step is 14 

f = d + L (e - c) 









3 11 

r o n 


6 2 



8 8 



3 3 



14 This correction can be carried out within the regular tabulation without writing 
out the numerical equation shown here. 




As a final check, let us get the ratio between the corresponding elements of 
f and e. 



= 29.84 

= 29.93 



= 29.88 

= 29.91 

Thus the lowest natural frequency of the system is 

< = ^x -L = 0.200^ 
1 k t 29.9 k t 

and the amplitude ratio of this mode is 

1 : 1.412 : 2.935 : 0.327 


If we are interested only in the natural frequency, we can stop the 
iteration process sooner and use Rayleigh's quotient. Suppose we stop 
at the vector a. 

a=[l 1.4 2.9 0.3] 

Rayleigh's quotient can be computed by the following tabulation: 


a C a 

" 4 



1.0 x 1.2 = 1.20 




1.4 x 2.1 = 2.94 



2.9 x 1.5 = 4.35 


0.3 x 0.6 = 0.18 

I C a = 8.67 x k 


a A a 




1.0 x 5.8 = 5.80 


4 1 

1.4 x 10.5 = 14.70 

z t 

1 2 


2.9 x 7.5 = 21.75 



3 x 4.6 = 0.96 

a A a = 43.2 x k t 

< = 



6k t 

= 0.201 ^ 

(d) determination of a second eigenvector 

To get other eigenvectors by iteration, we need some supplementary 
theorems. These are now discussed. 


Let the eigenvalues and the eigenvectors of a matrix L be 

a>i, A 2 , • • • , A n 

-(1) -(2) . . . r (n) 

The matrices al_ and L — pi, in which I is the identity matrix, have the 
same eigenvectors and their eigenvalues, respectively, are 

aA l9 aA 2 , • • • , <xX n 

a ± - & a 2 - n, • • • , (x n - p) 

The proof of this theorem is simply as follows: 

(aL) r<*> = a [L r«>] = oV™ 
(L - pi) p«) = L r«> - j8r<« = (A, - 0)r«> 

Next, let L and M be two matrices having the same set of eigenvectors. 
Let their eigenvalues be, respectively, 


X l5 / 2 ) ' 

' 5 *n 

The product matrix P, 

/"l, ^ 2 » • • 

* , /«, 

P = L M 

also has the same set of eigenvectors with eigenvalues 

^l/*l> *2^2» ' ' ' > ^n^n 

This theorem is true because 

P r<<> = L(M r (l '>) = L^r<« = A^r<*> 
From these two theorems we can deduce that the matrix 

P = L(^I - L) 
has the same eigenvectors as L, and its eigenvalues are 

0> A 2 (^i — ^2)5 ^3(^1 ~~ ^3)' " * " 5 ^nWi — A n ) 
This result gives us a way of evaluating eigenvalues, other than the largest 
one, by the same iteration process described. 

Let us assume that we are given a matrix L whose eigenvalues are known 
to be real positive numbers : 

X x > A 2 > A 3 • • • > X n > 

First find A x and r (1) by iteration. Afterwards, form the matrix 

P = LtfJE - L) 

which has the same set of eigenvectors as L. Then iterate P to get the 
eigenvector with the largest eigenvalue. This eigenvector cannot be the 
r (1) just found, since the eigenvalue of P corresponding to r (1) is zero 




and therefore cannot be the largest one. To which of the eigenvectors the 
iteration of P will converge cannot always be established beforehand. 
We have, however, a useful clue. If 

k > a, + a 3 ) 

it is easy to show that 

* A - h) > UK - h) > UK -h)-- 

In that case the iteration of P will yield the eigenvalue-eigenvector pair 

UK ~ h) and r< 2 > 
We can also show by theory of algebraic equations that the sum of the 
diagonal elements of L is equal to the sum of its eigenvalues. Or 

'n + h% + • • ■ + l nn = K + A 2 + A 3 + • • - + K 
Hence, knowing A l5 we know also the value of A 2 + A 3 + . . . + X n . 

Very often this sum is less than X x and the inequality 

/Ij !> /l 2 4* a 3 
is assured. 

Let us now return to our previous example. There we have found the 
largest eigenvalue A x of the matrix L. 

"8 9 3 r 

7 12 6 2 

7 18 18 8 

3 3 

A x = 30 

The sum of all four eigenvalues of L is 8+ 12+ 18 + 3 = 41. Hence 
A x is larger than the rest of the eigenvalues combined, and the largest 
eigenvalue of the matrix 15 

P = L(A X I - L) 


X 2 (X X — A 2 ) 

"8 9 3 1" 

"(30-8) -9 -3 -1 

p = 

7 12 6 2 

7 18 18 8 

_0 3 3_ 


-7 (30 - 12) -6 -2 
-7 -18 (30-18) -8 
0-3 (30 - 3) 

" 92 36 -45 -23" 

28 45 -27 -25 

-98 -63 63 29 

-21 -54 27 57_ 

15 It i 

5 not necessary to 1 

ise tl 

ie exact value of X x in the p 

rocedure that follows. 


Iteration of this matrix yields 




7< 2) = [1.000 0.565 -1.261 -0.670] 
A 2 (30 - A 2 ) = 184.5 
A 2 2 - 30A 2 + 184.5 =0 

30 - V900 - 738 


The other root of the quadratic equation has no meaning. To get accuracy 
as well as a check, we should go back to the original matrix and compute 

L r< 2 >. 



























The answer checks out, and the eigenvalue is A 2 = 8.63. 
To get another eigenvector, we can form the matrix Q, 

Q = L(30I - L)(8.63I - L) 

and repeat the process as before. 

We conclude this discussion with two remarks. 

(i) Although theoretically by iterating the matrix (^1 — L) we should 
get the eigenvector r (7l) having the smallest eigenvalue, it is usually not 
practical to do so because the first few eigenvalues of this matrix {X 1 — X n ), 
(A x — A 7 j_ 1 ), etc., are often close together so that convergence is very slow. 

(ii) The multiplication of two n x n square matrices represents as much 
computational labor as multiplying a square matrix to a column matrix 
n times. Therefore, for moderately large matrix iteration problems labor 
may be saved by not getting the product matrix P (or Q) first. Instead, 
we can iterate a trial vector by multiplying it alternately with (XJ. — L) 
and L. 

It is not even necessary to iterate with these two matrices alternately. 
One can be used more often than the other. The principle is as follows. 
Let the starting vector be 

■ d) 

v = ai r u > + a 2 r^> -\ 1- a n r 




After being operated on by (XJ. — L), the result is 

v' = 6 r<» + a^ - A 2 )r< 2 > + • • ■ + *„Oi - A B )r<«> 

Theoretically, e is zero, but because of unavoidable round-off errors as 
well as the error in the value of X x , v' still retains a small component in 
r (1) . Next, we operate v' with L i times: 

V v' = 6V (1) + * 2 V(*i - h)r (2) + • • • + *nKKK - K)r {n) 

By this time the coefficient of r (1) , that is eX-f, may become a fair amount. 
So we suppress it by operating again with (^1 — L) and then follow 
by operations with L. 

The sequence of operation is thus 

V(X x l - L) L V^l - L) v 

It is clear that the greater the precision in the value of X 1 previously obtained 
and the smaller the round-off errors the less often it is necessary to use the 
matrix (A X I — L). It is also true that if additional eigenvalues are to be 
found, regardless of the scheme adopted, the preceding eigenvalues must 
be determined with greater accuracy. The last remark applies also to the 
method discussed below. 

(e) determination of successive eigenvalues by successive 
reduction of the order of the problem 

When one of the modes of vibration is known, it is possible to reduce 
the order of the problem by one. The situation is exactly the same as that 
discussed in Art. 3.9, in which a semidefinite system is reduced to a positive- 
definite system by introducing the constraint relation (74) to suppress the 
zero mode. The procedure is best illustrated by an example. Take the 
eigenvalue problem we have just studied. After we have found the first 

H 1 * = [1.000 1.412 2.935 0.327] 

we introduce either of the constraint relations 

? C r<« = and FA r™ = 

This is equivalent to restricting the vibrations only in the remaining 
modes, which are all orthogonal to r (1) with respect to either C or A. 
Putting in numerical values, we have for this example 

(4 x 1.000 - 2 x 1.412>i + (-2 x 1.000 + 5x 1.412 - 1 x 2.935)r 2 

+ (-1 x 1.412 + 1 x 2.935)r 3 + (2 x 0.327)r 4 = 

r t = -1.810r 2 - 1.301/<3 - 0.531r 4 




By substituting this relation into the original problem, 



3 1" 





6 2 
18 8 


= A 


3 3 _ 



we will have four linear equations with three unknowns. One of these 
equations can be considered redundant and be left out. If we leave out 
the first of the equations, the following reduced system is obtained: 

^12 - (7 x 1.810) 6 - (7 x 1.301) 2 - (7 x 0.531)' 

18 -(7x1.810) 18 -(7x1.301) 8 - (7 x 0.531) 

- (0 x 1.810) 3 - (0 x 1.301) 3 - (0 x 0.531) 

This procedure can be generalized as follows. Find a constraint relation 
through the orthogonality property of eigenvectors and express it as 

r x = a 2 r 2 + a 3 r 3 + • • • + a n r n 

The problem is then to find the eigenvalues of the reduced matrix L': 

'22 H" '21 a 2 '23 + '21 a 3 ' ' ' '2n H~ '21 a n 

r ^ 2 ] r^ 2 " 

r 3 = 1 r 3 

_ r 4_ L r 4_ 

L' = 

'32 i '31 a 2 'as ~r «ai a 


! 31^-2 



'zn i '31 a « 


+ /»1«2 / n 3 + 4l«S 

'nrc T" 'nl*r!, 

The eigenvectors of L' have only (n — 1) elements, which are the same as 
the corresponding elements of (n — 1) eigenvectors of L. The remaining 
elements of the eigenvectors of L are obtained by the constraint relation 


So far, in this article and in the next, the discussion centers on the 
mathematical problem of determining the eigenvalues and eigenvectors of 
a square matrix. To connect this discussion with the solution of physical 
problems, the following remarks are presented to bring together pertinent 
results previously obtained. 

Assume that we are dealing with vibrations of systems in which damping 
forces can be neglected. For such systems the first thing of interest is the 
natural frequencies and modes of vibration. This we have shown to be an 
eigenvalue problem of the square matrix A -1 C or C^ 1 A. Since the 
numerical methods discussed always give the largest eigenvalue first 
and since in practice we are more interested in the lower frequencies, the 
matrix product C _1 A (hence C -1 ) has to be found. The matrix C -1 is 


formed by the influence coefficients between different pairs of generalized 
coordinates. If these coefficients can be computed directly, a tedious and 
error-producing inversion procedure can be eliminated. For systems 
such as beams and frames, carrying masses, the influence coefficients are 
directly obtainable from formulas in strength of material, whereas the 
elastic constants, which form the elastic matrix C, are not. On the other 
hand, there are systems whose potential energy expressions in terms of 
generalized coordinates can be readily obtained. For such systems it is 
more convenient to determine C first and then invert it to obtain C _1 . 

After the matrix product C -1 A is obtained, the methods described can 
be used to determine the natural frequencies and amplitude ratios of the 
system in free vibration. Ordinarily, only the first few modes are of 
practical interest. This is because of two facts. First, the frequencies of 
external disturbances are usually far below the higher natural frequencies. 
Second, the assumptions required in setting up the differential equations 
are often very unrealistic for the higher modes of vibration, although they 
may be justified for the lower modes. 

When the numerical procedure is terminated after the determination of, 
say, the first k modes, only the first k columns of the modal matrix R 
defined by (18) are known. By knowing the first k columns of R. we can 
compute the first k diagonal elements of the matrices M and K, since 

M = R A R and K=RCR 

In this way the first k equations represented by (46) can be obtained. By 
solving these equations for p 1 ,p 2 , • • • , p k anc * then assuming/?^ = p k _ 2 — 
. . . = p n = 0, we can get an approximate solution for q l9 ql, . . . ,q n in a 
forced vibration problem. This procedure is permissible when the fre- 
quency of the external force or forces is low in comparison with the higher 
natural frequencies of the system. For example, a system having 7 degrees 
of freedom is under an external sinusoidal force of 60 cycles per second. 
The first three natural frequencies were found to be 44, 185, and 602 
cycles per second. The system's approximate response to this external 
force can then be determined without finding all of its natural modes. 

3.12 Additional Theorems and Methods 

(a) enclosure theorem 

In many practical problems the generalized coordinates of the system 
are the displacements of the different inertia elements with respect to some 
fixed reference system. For such problems the kinetic energy expression 
contains only the squares of ^'s. 

T = Jfaitfi 2 + «22^ 2 2 + * * * + a nn q n 2 ) 


In other words, the matrix A is a diagonal with positive diagonal 

a {j = when i ^ j and a {j > when i = j 

In matrix iteration of the eigenvalue problem for such a system we have 
the so-called enclosure theorem, which can be stated as follows. Let 

L = A 1 C 


v' = Lv 

in which v is an arbitrary vector. Let the ratios between corresponding 
elements in v' and v be 

A l — > A 2 — > * ' ' j A n — 

Vl v 2 V n 

Let the largest of these ratios be A^ ax and the smallest, A f min . The theorem 
then asserts that between these two limits there is at least one eigenvalue 
of L. 16 

The conditions stated for the validity of this theorem are rather stringent. 
For example, it does not apply to the case 

L v = (C- 1 A) v = v' 

in which A is diagonal and C _1 is symmetric. However, these conditions 
are sufficient conditions. Although we have examples to show that the 
theorem is not valid for arbitrary L, it is suspected that it is valid under 
more liberal conditions, which at present are not known. Experience 
seems to indicate that when A.' maLX and A' min are reasonably close together they 
usually bracket an eigenvalue. 

(b) eigenvalue problem of a symmetrical matrix and 
symmetrization of a general eigenvalue problem 

The eigenvalue problem of a symmetrical matrix 

Sr = Ar (S = S) 

has many convenient features. One is that the enclosure theorem just 
discussed is valid; another is that Rayleigh's quotient of vectors during 
an iteration process is easily computed : 

vSv v v' 

R(y) = -=— = =- 

V V V V 

16 See Eigenwertaufgaben mit technischer Anwendung, by L. Collatz, Akademische 
Verlagsgesellschaft, Leipzig, 1949, p. 289. 


Still another is that the eigenvectors are mutually orthogonal in the 
ordinary geometrical sense; that is 

Among other things, this simplifies the procedure described in Art. 3.1 1(d). 
Moreover, if the eigenvectors are normalized in length, that is, by proper 
scaling of these vectors so that 

f«> r a) = i 

then the modal matrix formed by the r's will have the very desirable 
property that 

R R = I or R = R 1 

The truth of these statements can be readily verified. 

Now we want to show how a general free vibration problem can be 
converted into that of a symmetrical matrix. The idea is simply to convert 
one of the energy expressions into a sum of squares by a convenient linear 
transformation of variables. Suppose we choose the potential energy 
expression for this maneuver. 

V = iq C q 

First we find a matrix F such that 

FF= C 

This is always possible when C is symmetrical. Introduce a new set of 
coordinates represented by g. 

g=Fq or q = F 1 g 

This transformation changes the energy expression into 

K=iqCq = iqFFq = igg 

T= iq Aq = ^(F 1 A F"i) g = Jg S g 

S = F 1 A F 1 

can be shown to be symmetrical. The eigenvalue problem then becomes 

S r = Ar I = -i ■ 


Similarly, if we make 

FF = A and FSF=C 

we have a symmetrical eigenvalue problem with X = co 2 . 

At first glance it appears that in our effort to symmetrize the problem 




a great deal of additional labor of computation is introduced to defeat 
our original purpose. But this is not so, for by a proper choice of F the 
computational labor involved will be no more than that of getting C _1 A 
or A -1 C, and in many ways it is more convenient to manage. 
The matrix equation 

FF= C 

represents a system of n 2 linear simultaneous equations, of which n{n — l)/2 
repeat themselves because C is symmetrical. Therefore, the elements of F 
are not uniquely determined and n(n — l)/2 of the elements may take 
arbitrarily assigned values. Since the object is to find as simple an F as 
possible, we will assign zero as the value of these elements. The most 
advantageous way of placing these zeros turns out to be all at one side of 
the diagonal line. In other words, F is a triangular matrix. 

711 712 713 * ' * fin 
722 723 ' ' ' J2n 

f„ ••• f 3n 

F = 

0— /„„. 

The expansion of the matrix equation FF = C yields the following 
simple relations: 

Tii = c n 

7ll/l2 == C 12' J11J13 = C 13' ' ' ' 'Jlljln = C \n 

7 22 = C 22 — 712 

722/23 = C 23 — 712713' 7 227 24 = C 24 — 712714' 

7 33 == C 33 ~~ 713 ""7 23 

7 33J34 = C 34 ~~ 713714 ~ 7 23724' 7 337 35 = C 35 — 7 137 15 ~~ 7 23725' ' 

f 2 = C — f 2 — f '' 

J nn ^nn Jin Jin 


The salient feature of these equations is that the elements of F can be 
computed one at a time starting from the left to the right in each row and 
then row after row. A convenient tabulating schedule can be set up to 
handle this computation. 

After having found F, we will now find F _1 . Again a triangular matrix 
has the advantage that its inverse can be computed one element at a time 
in a simple way. This is in marked contrast to the determination of the 
inverse of an arbitrary matrix. As a matter of fact, there is a number of 
practical schemes for the determination of the inverse of a matrix based 
on the principle of splitting the matrix into two triangular matrices. 


Since f it = when i >/', the expansion of 

F 1 F = I 

is ./irVn = i. 

fll~fl2 + fl2~ J22 = 

/ll 7l3 +/l2 723 + /l3 7 33 = ^ 

/ll /in + /l2 72n + ' ' + fin fnn ~ 

/irVii = or / 21 -i = 

J22~~ J22 = 1» J22~J2Z + 7 23~ 7 33 = 

/«r x = o, /„-* = o, As" 1 = 
/■ _1 /" =1 

J nn J nn L 

It is seen that F -1 is also a triangular matrix, and its elements can be 
computed from F one at a time in a simple way. 

3.13 Chain Systems — Holzer's Method 

One way of obtaining the frequency equation of the vibration problem 
embodied in 

(Ca - a ii o^)r i = I (C - Aco 2 )r = 

is as follows. Set aside one of the n simultaneous equations represented 
by this matrix equation. Solve the remaining n — 1 equations by assigning 
an arbitrary value to one of the coordinate variables. The solution will be 
in terms of co 2 . By substituting this solution into the equation that has been 
set aside, we obtain an equation in co 2 , which is the frequency equation of 
the system. This method is usually too tedious. Under some special 
circumstances, however, the principle of this method can be applied to 
determine the natural modes. 

Consider the torsional vibration of a shaft carrying a number of disks. 
as shown in Fig. 72. The eigenvalue problem for this system is that of the 
following set of algebraic equations. [See Art. 2.9(a).] 

+ /jW 2 ^ = fc 18 (^ - </> 2 ) 

^12(^1 — ^2) + h c ° 2< t ) 2 ~ ^23(^2 — ^3) 
^23(^2 - </> 3 ) + V^Vs = ^34(^3 - 4 ) 

K-lJtn-1 ~ <f>n) + />Vi = 




Physically these equations describe how the torque of the shaft changes 
from one section to the next. We start from zero torque at the left and 
end up with zero torque at the right end. 

Holzer's method of solving this problem is as follows. Assume first 
a value for co 2 and assign the value 1 to </> l5 which can be arbitrary anyhow. 
From the first equation we then get ^ x — (/» 2 and (/> 2 . By putting these into 
the second equation, we obtain </> 2 — </> 3 and </> 3 , and so forth. When we 
reach the (n — l)th equation, we shall have all the values of the </>'s. The 
last equation then tells us whether or not our assumed value for co 2 is 
correct. By trial and error a value for co 2 will eventually be found that will 

: 23 



Figure 72 

make the left-hand side of the last equation vanish or become very small. 
This value is the square of a natural frequency, and the corresponding 
values of (/>!, <£ 2 , . . . , <j> n represent the elements of the corresponding modal 

To facilitate the setting up of a tabular method for the computations, 
the preceding set of equations are rewritten: 

^co 2 ^ + I 2 w 2 <f> 2 = ^23(^2 - <W 
I^j 2 ^ + / 2 co 2 (/> 2 + I s co 2 (/> 3 = k u ((j> 3 - 

/ lQ >Vi + Vo 2 <£ 2 + • ■ • + K^ n = 


The computation for each trial value of co 2 consists in filling out Table 1 . 
Columns A and F are first computed from the given data with an 
assumed value of co 2 . The rest of the places are filled in according to the 
sequence indicated. Item D n represents the left-hand side of the last 
equation, which is a function of the value of co 2 assumed beforehand. 
The idea is to determine the value of co 2 for which 

D n (co 2 ) = 

To start the trial-and-error procedure, we must have a reasonable estimate 
of the correct co 2 , lest the labor needed be prohibitive. To this end we 




first compute Rayleigh's quotient with a roughly estimated amplitude 
ratio </> x : <f> 2 : . . . : </> n . Since the system is semidefinite with a zero mode, it 
is necessary that this estimated ratio satisfies 

h<h + IJt + • ' • + h<f>n = 

Rayleigh's quotient obtained with this constraint will always be larger 
than the square of the lowest nonzero natural frequency of the system. 
It is therefore preferable to start our trial-and-error procedure with a 
value of co 2 somewhat lower than that of Rayleigh's quotient. 17 




Table 1 

CO 2 = 







Im 2 


Ioj 2 (f> 

^Ioj 2 cf> 

= k \<f> 

ZIoj 2 4> 



k ^ 

l x OJ 2 

hco 2 

I^ 2 

I n OJ 2 

B 1 -E 1 
B 2 — E 2 

A 1 B 1 
A 2 B 2 
AB 3 


C 3 + D 2 

D 1 F 1 

D,F 2 

\Jk 12 


After the first trial, it becomes necessary to have a way of knowing 
which way to correct the co 2 value for the second trial. In other words, 
we must know how D n varies with co 2 in the vicinity of the trial value. 
It was mentioned previously that D n (co 2 ) = is the frequency equation 18 
if co 2 were left in as a variable instead of an assumed numerical value. 
The value of D n is therefore bounded and is a continuous function of co 2 
for all finite and positive co 2 . A plot of D n versus co 2 must be of the form 
shown in Fig. 73. The curve crosses over the abscissae at n places, including 
the origin. These correspond to the natural frequencies. At the origin 
the curve must have a positive slope, since for a very small but positive 
o) 2 the sequence of computation must lead to a positive D n . Hence in our 

17 We are assuming that generally we want to find the lowest natural frequency first. 

18 This equation may differ from (16) only by a multiplication constant. 




ndicates that 

trial to locate the lowest nonzero frequency a positive D 
the trial value is too small and vice versa. 

In practical applications Holter's method is most widely used for 
analyzing the torsional vibration of the crankshaft of a multicylinder 
engine connected to a generator or other driven systems. In such appli- 
cations there are usually two or three large inertia elements representing 
generator rotor, flywheel, damper inertia, etc., and a number of small 

Figure 73 

inertia elements representing the pistons, connecting rods, and crankshaft 
mass of the cylinders. To obtain an estimate of the lower natural frequen- 
cies of such a system, we may simplify it by "lumping" the smaller moments 
of inertia with the larger ones near to them. Then the simplified system 
has only one or two nonzero modes whose frequencies can be determined 
quickly with methods described in the preceding chapters. These fre- 
quencies may now serve as the starting values for Holzer's method. 

Holzer's method may also be used alternately with Rayleigh's method. 
After a trial run Rayleigh's quotient for the amplitude ratio can be obtained 
by the following simple relation: 

o/2 = R(^ l9 </> 2 , • • • , </> n ) = 

Hk A(/> 2 _ 
2/^ 2 " 

M + D 2 E 2 + 

• • + D^E^W 

(^Q + B 2 C 2 + ■ ■ • + B„C„) 

in which the symbols refer to items in Table 1. If D n = 0, co' naturally is 
equal to co, and the correct frequency is obtained. If D n is small, co' is a 
better approximation than co and can be used for the next run of computa- 
tions by Holzer's method. If D n is large, a corrective procedure, which is 
described later, may be needed. 




We shall now work an example. Let the crankshaft of a 4-cylinder 
engine and the attached inertia elements be equivalent to the system shown 
in Fig. 74a. The values of the 7's and k's are 

I x = 150 

7 2 =20 

h = h = h = h = 2 

7 7 = 10 

/c 12 =l 

^23 = " 

^34 = ^45 = ^56 = ^ 

^67 = 4 

All are in their proper units, so that the unit of co will be 10 z radians per 



s 67 


I/= 150 

72 =24 

/,' = 14 


K 23 — i7 

Figure 74 

If we lump 7 3 and 7 4 with 7 2 and 7 5 and 7 6 with 7 7 , we have the approxi- 
mately equivalent system shown in Fig. 74b, in which 

// = 150 7 2 ' = 24 

h' = 14 

k' = 1 

^23 — 1,1,1.1.1 
613 + 3^3^4 



By substituting these into (53) of Art. 2.9(a), we obtain the following 
frequency equation : 

12/' 14 
F7 1+ 24 





This quadratic equation gives 

co 2 = 0.0262 or 0.1000 (10 3 rad/sec) 2 

We can now proceed to construct Holzer's table, Table 2, with co 2 = 

Table 2 

a> 2 = 0.0262 







Ico 2 


IC0 2 <f> 








































-0.007 = 

D 7 

Since the value of D n in our first trial is sufficiently small, we have the 
first natural frequency of the system; co^ 2 = 0.0262 x 10 6 . 

Of course, we cannot always be lucky enough to have the estimated 
answer so close to the true answer. For the purpose of illustration, let 
us now work the same problem with a much poorer initial estimate. 
Suppose by some scheme we obtain an initial estimate of co 2 = 0.0328. 
This estimate results in Table 3. 

Since D n = —1.084 < 0, we know that the trial co 2 is too large. We 
can decrease this value and try again. A better way would be to compute 
Rayleigh's quotient with the approximate modal vector represented by 
column B of Table 3. According to our previous discussion, 

D X E X + D 2 E 2 + ■ • • + D % E % 

CO * = CD* 

B 1 C 1 + B 2 C 2 + • • • + B 1 C 1 



x 0.0328 = 0.0261 

Thus we see that Rayleigh's quotient of the modal vector obtained from a 
roughly estimated value of co 2 gives remarkably accurate results. 




The modal vector obtained from column B of a first trial generally does 
not satisfy the constraint relationship 2/c/> = 0, since D n ^ 0. It therefore 
contains a rigid-body motion. If D n is not too large, we may disregard 
this discrepancy. If D n is moderately large, a correction may be necessary. 



w 2 = 0.0328 







/CO 2 


Io> 2 <t> 









































-1.084 = 

D 7 

The correction consists of deducting from the denominator of Rayleigh's 
quotient a term representing the kinetic energy of the zero mode. This is 
permissible because of the orthogonality relation of the modal vectors. 
The correction term is the first term of the denominator of (39). For the 
present problem it is 

€ = (A + h H T- 4) <£mean 

in which 

, _ /A + / 2 < / > 2 + ••• + /„<£„ 

*n*an ~ ^ + ^ + . . . + j 

In terms of quantities in the tabulated computation, we can see easily that 


(A 1 + A t + ~' + A n )co< 

Applying this formula to the example of Table 3, 

1.084 2 _ 0.164 
€ ~ 7.162co 2 ~ co 2 


Rayleigh's quotient corrected is then 




(35.74 - 0.16) 


although in this example this correction is too small to warrant the effort. 
To obtain the second nonzero mode, we begin by assuming oj 2 = 0.1000. 
The computations are tabulated as Tables 4 and 5. 

Table 4 

co 2 =0.1000 














































-7.522 = 

= D i 

ZDE = 586 
o>' 2 = 

ZBC = 526 D 2 \ 

^—^(0.1000) =0.11 

S/l = 3 


By this time it appears very probable that the iteration process will 
approach the answer from below. Rayleigh's quotient in Table 5 is most 
likely to be very near but slightly lower than the true answer. The first 
trial results in a 13 per cent increase and the second trial, a 5 per cent 
increase. For the final trial we will use an "anticipated" answer, which is 
about 2 per cent larger. 

This time, D 1 is positive but very small, so that we know the true answer 
is slightly below 0.121. 



Table 5 


CO 2 = 















































-3.28 = 




DE = 964 


/ 2 = X 


25C = 914; 
0.113 =0.119 

Table 6 

co 2 =0.119 x 1.02 =0.121 














































+0.25 = D 1 




3.14 Electrical Analog of Mechanical Systems 
and Electromechanical Systems 

(a) electrical analog 

We have emphasized on different occasions that vibratory systems may 
have entirely different physical appearances, yet their motions are governed 
by common mathematical equations, so that it is not always necessary to 
analyze each individual system having a different make-up. We may go 
one step further and observe that the differential equations of motion 

C R L 

— II — vwv\ — nmrr^-. 

(J eft) 

Figure 75 

which describe vibratory phenomena also govern the behavior of many 
other types of systems. These systems are called analogous systems, and 
they also come under the same treatment. The analogous systems most 
closely related to mechanical vibratory systems are electrical networks. 
Except for a different placing of emphasis, network theory is almost 
indistinguishable from vibration theory. Historically, these theories have 
always been borrowing results from each other. 

Consider the simple circuit shown in Fig. 75. It consists of a resistance 
R, an inductance L, a capacitance C, and a time-dependent voltage source 
e(t) connected in series. If i(t) denotes the loop current, Kirchhoff's law 
will lead to the following integro-differential equation: 

Li + Ri + 



dt = e(t) 


The indefinite integral above contains an integration constant to be 
determined by the initial condition. By denning 




we can write this integro-differential equation as 
Lq + Rq + -q = e(t) 






This equation is then the same as that for the forced vibration of a single- 
degree-freedom system. 

rnq + cq + kq =f{t) 

The electrical circuit in Fig. 75 is thus an analogous system of a spring- 
mass-damper system in the sense that a time-dependent voltage source will 
produce a current in the circuit in an analogous way as a time-dependent 
force gives a velocity to the mass of a spring-mass-damper system. This 
analogy is often called the direct analogy or the force-voltage analogy. 



Figure 76 

A different electrical analog of the same mechanical system is the circuit 
shown in Fig. 76. There we have a capacitance, a resistance, and an 
inductance connected in parallel to a time-dependent current source i(t). 
Let v(t) be the voltage drop across the three parallel elements. The branch 
currents flowing in these elements are, respectively, 

v J If 

i c = Cv i R = — and i L = — v dt 

Since the sum of the branch currents must be equal to the current from the 
current source, we have 

1 1 f 

Cv + - v + - \v dt = i(t) (iv) 

This integro-differential equation has exactly the same mathematical form 
as the one before, and because the indefinite integral contains an arbitrary 
constant this integro-differential equation can also be changed into a 
differential equation of the second order by defining a function 19 q{t) as 


q(t) = \v(t)dl 

cq + p + \i = m 

19 This definition is intended to have mathematical meaning only. Although the 
integral \i dt previously represented the physical quantity, electrical charge, this 
physical meaning is not essential to our discussion. 


This analogy is called an indirect analogy or & force-current analogy. Here 
the voltage response to a current source is analogous to the velocity 
response of a mechanical system to an applied force. 

The foregoing simple examples show that when we speak of analogous 
systems we must have in mind some specific phenomena with which the 
analogy is to be drawn. In the present case the phenomena in question 
are the signal-response relationships between analogous pairs of physical 

We must realize also that both the mechanical system and the electrical 
systems used for illustrations are merely models of other systems to which 
they are analytically equivalent. Hence the analogy exists not only between 
two specific systems but between two classes of equivalent systems. 

An electrical analog to a mechanical system, which consists of a simple 
arrangement of springs, mass elements, and dampers, can be conceived 
by inspection without writing the differential equations of motion. There 
are rules available whereby an analogous circuit can be drawn simply by 
replacing mechanical elements with appropriate electrical elements and 
mechanical arrangements with analogous electrical arrangements. These 
rules permit a person who is more familiar with one type of system to 
analyze the other type with which he is not as familiar. It is, however, not 
our intention to analyze a mechanical system by converting it into a 
network problem or to analyze an electrical circuit by vibration theory. 
Our purpose of studying analogy is partly to create an awareness of its 
existence and partly to enable us to build electrical analogs for complex 
mechanical systems whose signal-response relations can then be determined 
experimentally on the analog. (It can be readily appreciated that experi- 
ments on electrical circuits are usually more economical than those on 
mechanical models.) Therefore, in the following discussion it is assumed 
that we know how to write the governing equations for both types 
of systems when they are needed for the construction of analogous 

Let us now study an example of a mechanical system having two 
degrees of freedom, as shown in Fig. 11a. The differential equations of 
motion are 

WiA + c xl x Y + k u x ± + c 12 (x x - x 2 ) + k 12 (x 1 — x 2 ) = f x (t) 

^22^2 ' ^22^2 • ^22^2 T ^12V^2 X l) ' ^12\ X 2 X l' =z j2\") 

The two electrical analogs of this system are shown in Figs, lib and 77c. 
The reader can verify that by applying Kirchhoff's law to the two loops 
in Fig. lib the loop currents are governed by the same set of equations, 
and by summing up the branch currents at the two nodes in Fig. 77c the 



nodal voltage are also governed by the same equations, 
tables help to describe the analogous relationships: 


The following 

Mechanical System 

D'Alembert's principle 
Degrees of freedom 

Generalized velocity 
Generalized force 

Inertia element 
Damping element 
Elastic element 
Coupling elements 

Stationary frame 


Kirchhoff's law 

Loop currents 
Voltage source 

1 /capacitance 
Elements common 
to two loops 

Continuity law 
Nodes (not including 

datum node) 
Nodal voltage 
Current source 

1 /inductance 
Elements between the 

Datum node 

The electrical analogs of mechanical systems, which consist of masses, 
springs, and dampers connected in a simple way, can be constructed 
according to the analogous relations given in the foregoing tables. For 
others, in which the couplings among the different degrees of freedom are 
not obvious by inspection, it is necessary that the differential equations of 
motion be obtained first. Take for instance the system shown in Fig. 53 
and studied in Art. 2.11. There are no ready-made rules by which we can 
construct the electrical analogs for such a system without first writing 
down the differential equations of motion; these equations were found 
to be 

(M + m)q\ + Kq 1 — m cos 0q 2 — Kq 2 = 

in which 

— m cos Bq x — Kq x + mq 2 + (K + k)q 2 = 
q Y — X and q 2 = x 

For the purpose of constructing electrical analogs we rearrange the 
equations to read 

(M + m — m cos 6)q\ + m cos d(q\ — q 2 ) + K(q 1 — q^ = 

(m — m cos 0)q 2 + kq 2 + m cos 6(q 2 — q x ) + K(q 2 — q x ) = 

From this set of equations we deduce that the two electrical analogs are 
as shown in Fig. 78. 














® cz 



^22 ^ 

v/VVV\A— | 

fel1 dn m 22 ^2: 

r^WHh^rK^Mh J vVrn 

i x-^ (uah 




M " 12 £w) 

V-A^Current Jh^ 

v = 

Figure 77 

(Datum node) 

For a more complex example let us consider a mechanical system whose 
motion is described by 

Aq + Bq + Cq = f 

x 10- 

From the signs of the off-diagonal elements of these matrices we observe 
that in a force-voltage analog for this system the loop currents must be 

r 10 5 -ii 

[9 4-3" 

[21 3 


A = 

5 8 2 

B = 

4 12 6 

C = 

3 17 


-12 7 

-3 6 11 

-11 4 





arranged to flow in such a way that there will be positive couplings between 
loops 1 and 2 and loops 2 and 3 and negative couplings between loops 1 
and 3. The arrangement is shown in Fig. 79. The values for the coupling 
inductances, resistances, and capacitances are given by the absolute values 
of the appropriate off-diagonal elements of the matrices A, B, and C. 


(M + m - m cos 6) k (m-m cos 6) 



m cos 6 




v x (t) 



m cos 6 




v 2 (t) 

— o 






Figure 78 

Additional electrical elements are then placed in the loops to make the 
total inductance, resistance, and capacitance in each loop equal to the 
values of the appropriate diagonal elements of the matrices. Voltage 
sources are placed in each loop to fit the column matrix f. The polarities 
shown in Fig. 79 are for positive values of the elements in f. 

The construction of a force-current analog of the same system follows 
the same idea. It is left to the reader as an exercise. (See Exercise 3.21.) 

The types of analog circuits described so far contain only passive circuit 
elements, which are capacitances, resistances, and self-inductances with 





positive values. If these are the only kinds of circuit elements that may be 
used, it is not always possible to construct analog circuits from a set of 
differential equations obtained from a real mechanical system because to 
do so may require circuit elements with negative values. On the other 
hand, there are electronic devices that can be made to behave like negative 

Cl3 #13 Lis 


L>23 #23 ^23 


x 12 = 5 

^13= 1 

L 23 = 2 

X n = 10-5-1=4 
R n =9-4-3=2 

C u = - 10 / - = > > !(> 

C 12 = \ X 10- 6 
C 13 = A x 10- 6 
,c, = | x 10- 6 

x 29 


Roo = 12 - 6 - 4 = 2 



3 - 11 



Figure 79 

= 10- 

/?!<> = 4 

/? 13 = 3 
/? 23 = 6 

•L 33 = 7-2-1=4 
/?„ = 1 1 -6-3=2 

10- 6 

k 33 


(short circuit) 

capacitances, resistances, and inductances. These devices contain energy 
sources in themselves and therefore are not passive elements. They are 
commonly used in analog computers, which are nothing but electrical 
analogs designed with a greater flexibility of operation. 20 


The analogy between vibratory mechanical systems and electrical 
circuits is a particularly intimate one because in both types of systems the 
phenomenon of interest is the transformation of energy forms viewed 

20 For a brief account of analog computers see Engineering Systems Analysis by R. L. 
Sutherland, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958, Chapters 7 
and 8. 


against the background of time. In a pure mechanical system the funda- 
mental relation is 

-(T + V) + P= W 


where P is the time-rate of dissipation of mechanical energy, 21 which is 
usually in the form of heat, and W is the time-rate of work done by 
external forces. Similarly, in an electrical system we have the energy 

-(S+U) + P=W (v) 


where S = electric energy 
U = magnetic energy 
P = time-rate of (heat) dissipation 
W = input from external power source 

Consider the simple circuit shown in Fig. 75 as an example. In terms of 
the loop current i the energy expressions are 

S = - q - U = -Li 2 P=Ri 2 and W = ei 
2 C 2 

where q is defined by (ii). If we substitute these expressions into the energy 
equation (v) and cancel out an i, the result is the equation from Kirchhoff's 
law (i): 

Li + Ri + -\idt = e(t) 

Now take the circuit shown in Fig. 76. In terms of the variable v, in this 
case the energy expressions are 

= 1 Cv 2 U = — I I v dt I P = - and W = iu 

2 2L \J f R 

and the energy relationship leads to the nodal current equation (iv): 

1 1 r 
cv + jv + -\vdt= i(t) 

21 The function P is the same as 2F, F being Rayleigh's function described in Art. 3.8. 


It is well known that mechanical energies and electromagnetic energies 
from the point of view of thermodynamics are reversibly convertible. 
Hence a mechanical system may be coupled to an electrical system to form 
a so-called electromechanical system. The first law of thermodynamics 
for such a composite system can be written 

[ T + V+ S+U] + P=W (vi) 


where P becomes the heat dissipation and W becomes the work input. 
The term "work" is to be interpreted in the general thermodynamic sense, 
which includes energy input by virtue of Newtonian forces as well as 
electromotive forces. The generalized coordinates for an electro- 
mechanical system consist of geometrical variables (displacements, angles, 
etc.) for the mechanical part and electrical variables (loop currents and 
nodal voltages) for the electrical part. Where the two parts are coupled, 
energy conversion takes place and imposes a relation between a pair of 
variables, one mechanical and the other electrical. This relationship 
serves as an equation of constraint, which describes the dependence of 
one on the other. 

An energy transfer between a mechanical phenomenon and an electro- 
magnetic phenomenon can take place in a number of ways. The most 
important is that which takes place when a current-carrying electrical 
conductor moves in a magnetic field. Take, for example, the simple case 
of the moving coil instrument studied in Art. 1.15. The basic construction 
can be schematically represented by Fig. 80. 

If we let = angular deflection of the meter coil 
/ = current flowing in the coil 
J = moment of inertia of the coil 
k = constant of the restoring spring 
R r/ = resistance of the coil 
R = resistance external to the coil 
e = externally impressed emf 

the energy expressions are then 

T = \Jd 2 V = \kd 2 

P = i\R + R g ) and W = ei 




We assume that the capacitance and the self-inductance of the coil are 
negligible and so is the viscous damping. By substituting these into the 
energy equation (vi), we have 

(j6 + k0)0 + i\R + R g ) = ei 


This equation contains both the geometrical variable and the electrical 
variable /. These, however, are related by an equation of constraint, which 

Flexible leads 


Figure 80 

may be found by the following reasoning, based on Faraday's law. As 
the coil turns, a back emf e b is produced, which is proportional to the 
angular velocity 0. 

e h = BO (viii) 

in which B is determined by the field strength of the pole pieces 22 and the 
geometry of the coil design. By applying KirchhofTs law to the loop, 
the relationship between and i is then 

e - e h = e - BO = i(R + R g ) (ix) 

By eliminating the variable i between (vii) and (ix), we obtain finally 

J6 + 

B 2 

6 + kO 


R+ R g 

R + R g 



22 It is generally the case that the magnetic field produced by the pole pieces is so 
strong and the coil so designed that the field strength is undisturbed by the current 
flowing in the coil. 


As it was pointed out before, the dependence relation between 6 and i 
originates from the process of converting mechanical energy into electrical 
energy, or vice versa. From this point of view we see that if the torque 
produced in the coil by the interaction between the current and the 
magnetic field is denoted by T b the rate of energy conversion is given by 

Tfi = e b i (xi) 

Hence the torque per unit current flowing in the coil is 

— = "J = B ( X1 

I U 

The dynamic equilibrium equation of the coil is 

JS + kd = T b = Bi (xiii) 

Elimination of i between (xiii) and (ix) results again in (x). Incidentally, 
(x) is the same equation as (i) of Art. 1.15 because of (xii). 

Although a moving coil instrument is a very simple electromechanical 
system, the analysis of its operation illustrates adequately the principles 
available for studying the class of system in which the conversion of 
mechanical energies into electrical energy, or vice versa, is effected by the 
motion of a conductor in a magnetic field. There are other types of 
conversion mechanisms which cannot be discussed here. Most of these 
involve highly nonlinear conversion relationships. 


3.1. A bent wire is contained in a vertical plane. Its shape is described by 

y = ax 2 for x > 

(R - yf = /?2 _ r 2 for x < 

A small ring slides on the wire without friction. What must the relation be 
between a and R so that the system may be approximated by a linear system for 
a small interval of x near x = 0? 

3.2. Write the energy expressions for the systems shown, using as generalized 
coordinates the quantities indicated. How are these expressions "linearized" 
for small oscillations? (The correct word is perhaps "bilinearized, ,, but it is 



Exercise 3.2 

3.3. Rewrite the following matrix equations first in tensor notation, then in 
longhand for n = 3. 

A f + ag = h g A B f = /. 

3.4. Rewrite the following equations in matrix notation: 

3.5. Write out in longhand the determinant in (16). 

3.6. Illustrate equations (12) to (16) with a numerical example using 

C = 

A = 


7 11 


19 7 


7 5_ 

14 3 
3 6 
-2 1 


2 J 

x 10 3 

Solve the frequency equation obtained and determine the modal vector corre- 
sponding to each frequency. 

3.7. To which of the equations in Chapter 2 are the following equations in 
this chapter equivalent: (12), (13), (14), (15), (16), (17), (19), (20) and (21)? 



3.8. By matrix iteration, determine the smallest eigenvalue and the associated 
eigenvector of the problem in which 

A = 


1.21 3.12 - 

ri3 3 



4.60 2.57 

C = 

3 9 



2.57 6.95 

7 4 


x 10 4 


(a) Find a second eigenvector for Exercise 3.8 by the method described in 
Art. 3.11(d). 

(b) Reduce the order of the problem of Exercise 3.8 to n = 2 by the method 
described in Art. 3.11(e). 

3.10. Show that a set of mutually orthogonal vectors must be linearly inde- 
pendent. That is 

o^r* 1 ' + a 2 r (2) , • • • , oc n r in) ^ 

unless a x = a 2 , . . . 
3.11. Prove that 

<2(u) > ^ 

by the fact that both the denominator and the numerator in (36) or (38) must be 
positive for arbitrary u. 

Hint. Show first that the elements of the diagonal matrix M are always non- 

3.12. Derive equations (47) and (48) in Art. 2.8. 

3.13. Set up the differential equations of motion for the beam shown. 




Exercise 3.13 

3.14. Find the steady-state solution of 

A q + C q = ie ia} f l 
in which A and C are given in Exercise 3.6, and 

f - [3, 4, 5] o>, = 20 
(a) by method discussed in Art. 3.7(b), (b) by methods discussed in Art. 3.7(c). 



3.15. Find C for the system shown. Does C _1 exist for this system? Can 
any of the influence coefficients be defined ? 


3.1-6. Given a symmetrical matrix A and a set of linearly independent vectors 

(a) Show that the set of vectors v (1) , v <2) , . . . , v (n) defined below are mutually 
orthogonal with respect to A. 

>(1> •.<!) 

(2) _ r <2) _ 

(3) _ -<3) 

v (1 >A r (2) 

v ,2) A r (3 ' 

v °' = r 

An) _ „(n) 

v (1) A r (3; 

v (2) A v (2) v (1) Av (1) 


y (n-l)^ r (n) 
;(n-l)j^ v (n-l) 


v (1) A r (n) 
v (1) Av (1) 


(b) Show that if the r's satisfy 

Cr=AAr A^O 
the v's defined are also mutually orthogonal with respect to C. 
3.17. Verify the statement following (52). 



3.18. Make up an eigenvalue problem for which the answer is 


r i I 

r i ~ 

r i 


r (2) _ 



r (3) _ 


I«(4) _ 




1 . 


X l = 1 A 2 = 4 A 3 = 8 A 4 = 20 
Note: The answer will not be unique. 

3.19. Find the lowest two natural frequencies by Holzer's method for the 
following torsional system 














lb-ft-sec 2 













3.20. Draw the force-voltage and the force-current analogs of the mechanical 
systems shown. 






Vibration of Elastic Bodies 

4.0 Introduction 

Vibratory systems having a finite number of degrees of freedom are 
only idealized models of certain real systems. The essential property 
of such systems is that the inertia forces are those due to a finite number 
of rigid bodies, and the potential and dissipative forces are the interactions 
between pairs of these rigid bodies. We have, therefore, rigid bodies, 
which can store 1 only kinetic energy, and weightless bodies and force 
fields, which can store only potential energy. Real systems consist of real 
bodies that are neither perfectly rigid nor completely without inertia, 
but they may be considered approximately one way or the other under 
certain circumstances. 

The validity of such simplifying approximations depends upon the 
relative rigidity and amount of mass in the different parts of the system 
as well as the nature of the vibratory motions being considered. For 
instance, in order for a simple spring-mass system to be considered a 
single-degree-freedom system, the mass must be relatively rigid and heavy 
in comparison with the spring, and the motion must be such that the 
amplitude of the mass is of the same order of magnitude as that of any 
portion of the spring. The last stipulation is necessary because the system 
may be set into high frequency vibrations by "stroking" the spring. In 
such vibrations the mass remains nearly stationary, while the different 
parts of the spring vibrate, and the system can no longer be considered as 
having only one degree of freedom. 

1 The concept of energy stored in bodies is sometimes convenient but cannot bear 
close scrutiny. The meaning of the word "stored" is not to be taken too literally in the 
present context. 



In this chapter we discuss the vibration of elastic bodies, which cannot 
be considered as rigid or as without inertia. The configuration of the 
systems containing such bodies thus cannot be described by a finite number 
of generalized coordinates. The analysis of these systems is naturally 
more complex. The increased complexities, however, lie mostly with the 
mechanics of elastic deformation (theory of elasticity) and the mathematics 
of solving boundary-value problems. There are no general methods in 
dealing with these two aspects of the problem. On the other hand, the 
conceptual matters related to theory of vibration can be discussed by a 
general treatment. This treatment we shall emphasize. Complete solutions 
for a few simple problems are carried out as illustrations. With these 
examples it is hoped that the reader will have enough understanding to 
analyze more involved problems, once he has also acquired the necessary 
knowledge of the theory of elasticity and the necessary mathematical 

4.1 Coordinates and Constraints 

In order to describe the configuration of an elastic body, it is necessary 
to specify the spatial location of every mass particle in the body. The 
most direct method of description is then to use the displacement of the 
particles as the coordinates. But the so-called mass particles in an elastic 
body are not discrete. They have a continuous distribution in space. 2 
Therefore, it is not possible to label them by integer indices or to assign 
finite values for their masses. 

Let x, y, and z be the Cartesian coordinates of a generic point P in an 
elastic body in static equilibrium. Let p(x, y, z) be the mass density at P. 
A generic mass element dm = p(x, y, z) dV then plays the role of a mass 
particle. Let w, v, and w be the three components of the displacement of 
this mass element from its equilibrium location. These components are 
both space and time dependent. 

u = u(x, «/, z, t) 
v = v(x, y, z, t) 
w = w(x, y, z, t) 

These displacement functions 3 can be considered as the coordinates of the 
system. The space variables x, y, and z are used to designate the particular 

2 We are looking at things macroscopically, not microscopically. 

3 Those who have studied fluid dynamics should note that they furnish essentially 
the Lagrangian description of the motion of a deformable body. 


element being considered. These variables play the role that was played 
by subscripts before. 

Since there are infinitely many combinations of x, y, and z, each corre- 
sponding to a mass element, there are infinitely many degrees of freedom in 
the vibration of an elastic body. At first glance it appears that the number 
of degrees of freedom is not only infinite but noncountable, since the mass 
elements in a continuous medium are not countable. 4 This, we shall see, 
is not the case unless the body is infinite in extent. The question of 
countability is of interest because we have seen that with each degree of 
freedom there is associated a natural frequency. If the degrees of freedom 
are countable, the natural frequencies will be discrete; that is, we can list 
them in an infinite sequence m v co 2 , . . . , co„, .... 

The coordinates u, v, and w, however, are not completely independent. 
They are subject to certain constraints. At the boundary where the body 
comes into contact with its support or with another medium the displace- 
ment functions are often subjected to certain restrictions, which are called 
boundary conditions. In the interior of the body other types of restrictions 
prevail, such as the requirement that the displacement function be con- 
tinuous and differentiable a certain number of times with respect to space 
variables. Restrictions also arise out of assumptions. For instance, in 
specifying the deformed shape of a slender beam, we may assume that the 
points on a given cross section move together as a rigid plane. 

Because of the different kinds of constraints described it is often possible 
to express a displacement function as the sum of an infinite series of given 
functions. For instance, in describing the shape of the elastic curve of a 
deflected beam simply supported at the ends, we may use the trigonometric 

™ nnx 

w = 2 q„ sin — 

n = \ L 

By specifying a set of values for the coefficients q lf q 2 , ■ . . , q n which 

are infinite but countable, the configuration of the beam is uniquely 
specified. Hence the set of </'s may be considered as the generalized 
coordinates of the system. This method of describing the configurations 
of elastic bodies is important because it allows us to borrow the results 
from the analysis in Chapter 3. 

4 By "countable" or "denumerable" we mean that the set of things can be put into 
one-to-one. correspondence with integers 0, 1, 2, 3, ... . It can be shown, for instance, 
that rational numbers are countable, whereas the number of points on a line segment is 


4.2 Formulation of a Problem by Differential Equation 

The direct application of Newton's second law to a mass element 
results in 

P-^dV=df x 

P^ dV = d fy 

P 1 f 2 dV=df z (1) 

in which df x , df y , and df z are elemental forces acting on the elemental 
volume dV. In free vibrations these are elastic forces caused by local 
deformations of the body. They are related to u, v, and w by the laws of 
elastic deformation, which relate forces to stresses, stresses to strains, and 
strains to displacements. General equations are available from theory of 
elasticity to express these forces in terms of displacements. 5 Since such 
general equations are seldom solved in practice, we shall dispense with the 
exact expressions here. It is sufficient to say that the equations are of the 

B 2 u 


P-*72 = L *( w ' y > w ) 

P^79 = L <>> v > w ) ( 2 ) 

in which the L's are linear differential operators involving partial deriva- 
tives with respect to space variables of different orders. 

■ / d d d i \ 

L = L — -, — -, — -, elastic constants 

\ox ay dz i 

It is to be noted that time t does not enter explicitly at the right-hand 

The partial differential equations so obtained are merely statements of 
Newton's law of motion and Hooke's law of elastic deformation, with 
certain references to the mass distribution and elastic property of the 
material that makes up the elastic body. These equations are, however, 

5 These equations are associated with the name of Navier. 


not specific enough to define a vibration problem. They must be supple- 
mented by other conditions. The first set of conditions is called the 
boundary conditions, which describe how the elastic body is supported and 
how it comes into contact with other media along its boundary. Differen- 
tial equations of motion, together with a set of boundary conditions, 
constitute a boundary-value problem. Such a problem specifies the types of 
vibratory motions a given elastic system may have. It is often the problem 
of practical interest. If, however, the problem is to find out how a given 
body actually vibrates, we shall also have to know how it was set into 
vibration to begin with. This information is furnished by so-called 
initial conditions, which specify the displacement and velocity distributions 
at a certain instant / = t . 

Let us now illustrate the discussion with a few examples. 

(a) longitudinal vibration of a slender bar 

and mathematically analogous vibration problems 

During longitudinal vibration each elemental length of the bar undergoes 
compression and extension alternately. If the bar is relatively slender, the 
inertia forces in the transverse directions, due to the contraction and 
expansion of the cross sections, are negligible. The internal forces in the 
bar are then essentially axial. The stress-displacement relationship in a 
bar subjected only to axial forces is given by Hooke's law as 

(y x = Ee x = E— (3) 


in which the x-axis is parallel to the axis of the bar, and a, e, and E are 

the stress, strain, and Young's modulus, respectively (see Fig. 81). 

Consider now the equilibrium condition for an elemental length of the 

bar between two cross sections spaced dx apart. Application of Newton's 

second law results in 09 ~ , ~ N 

o l u o I ou\ 

<> a w* = tA ea tJ (4) 

For bars with uniform cross sections the equation simplifies into 

d 2 u_Ed 2 u 
dt 2 ~~ P dx~ 2 

Since there is only one space variable, namely x, the mathematical 
model of the bar has only two boundary points corresponding to the two 
ends of the bar. 6 Various boundary conditions are possible. 

6 The boundary of the physical body itself naturally is a surface having an infinite 
number of points. The mathematical model reduces the bar into a one-dimensional 


At a fixed end the displacement is zero. 

u = 
At a free end the stress is zero. 


At an end at which a mass M is attached the inertia force of M and the 
elastic force at the end of the bar must be in equilibrium. 

d 2 « du 

tit 1 ox 

— -^ u(x, t) 



°x A n ff xA+-fe (<J x A)dx 


Figure 81 

At an end at which a spring of constant K is attached, the two elastic 
forces must be in equilibrium. 

Ku + EA— = 


Each of these boundary conditions, when valid, is supposed to hold 
only for one particular value of x but for all values of t. 

All of these boundary conditions are linear homogeneous because they 
contain only first power terms of the dependent variable wand its derivatives. 
The linear differential equation (4), and a set of linear boundary conditions 
together make the problem linear, and the superposition theorem holds; 
it is easy to see that if u x (x, t) and u 2 (x, t) are two solutions to the boundary- 
value problem, their linear combination c x u x + c 2 u 2 is also a solution. 

The boundary condition of the first type u = is described by the term 
"imposed" or "geometrical ." The other types are called natural or dynamical 
boundary conditions. The significance of this differentiation is discussed 
in Art. 4.7(b). 




The initial conditions needed to specify a particular solution of the 
vibration problem are of the form 




These conditions are linear but not homogeneous, since the right-hand 
sides are not zero. The statement of the superposition theorem as applied 
to an initial-value problem appears in Art. 1.10. 



u 2 

/\/\/\/v^/\/\/\/v4k/\/\/\/v^ #vW\A#y\/\/\/ 

"n + l 

m x 

m 2 

m 3 

m n _i 

Figure 82 

It is a worthwhile digression to compare the formulation of this problem 
with that of a similar problem in which the mass distribution is not 

continuous. Consider a system made up of n masses, m l5 m 2 m n , 

connected by n + 1 springs, k , k l9 k 2 , . . . , k n , into a continuous chain. 
(Fig. 82.) If all the masses are constrained to move along a straight 
line, with u lt u 2 , . . . , u n as their respective displacements, the equation of 
motion for the rth mass is 

d 2 u- 
Mi -^ = ki(u i+1 - u t ) - k^iu, - u^) 

1,2, ••-,/! 

In writing this general expression, u and u n ^ x are taken to be the displace- 
ments of the outer ends of the two end springs. To use the notation of 
difference equation, we set 

Aw 2 = u i+1 - u t 
Am,_i = u . _ Uil 

Mk, &u t ) = k t Au r - ki_ x Aw,_ x 
Hence the general expression becomes 

m/-^ = Wt Au t ) i=l,2,--,n 


If all the k's are equal, we have 
d 2 u, k 



dt 2 m, 

A 2 w ? . 

/= 1,2, 

These difference equations are "images" of the partial differential equations 
(4) and (5). 

The description of this system, as given by the difference equations, is 
not complete without some reference to the two end springs which terminate 

the chain. For instance, if one end of k n is fastened to a fixed point, we 
must also specify that 

u n <* = 

and, if one end of k n is completely free, 

u m ^ — u„ = Aw, 

These conditions correspond to the boundary conditions of the partial 
differential equation. 

Two other vibration phenomena have the same governing differential 
equation as the longitudinal vibration of a bar. The first is the torsional 
vibration of a shaft with a circular cross section. Let the axis of the shaft 
be in the ^-direction and the rotation of a generic shaft cross section be 
denoted by 

6 = 6(x, t) 
The torsion formula states 

30 __ M 
dx JG 




in which M is the torque, G the modulus of rigidity, and J the polar 
moment of inertia of the cross section. On the other hand, Newton's law 
of motion leads to 

d 2 6 _ dM 

P J ^2 

Upon elimination of M, the resulting equation is 

d 2 6 

P J 

dt 2 

dx \ dx] 

— H dx 








dt 2 

Figure 84 

For shafts with uniform cross sections 

d 2 e 
di 2 

Gd 2 d 
p dx 2 


Except for interpretation of the symbols, these equations are identical 
to (4) and (5). 

The second analogous phenomenon is the transverse vibration of a 
thin stretched string. Assume that the vibration is constrained within 
the #2/-plane. The configuration of the string during vibration is described 
by (Fig. 83): 

v = v{x, t) 


with v = being the equilibrium configuration. The force-acceleration 
relationship of an elemental length of the string in the ^/-direction is 

d 2 v 
P A fai dx = dS v 

in which S y is the ^-component of the tension S existing in the string. 
Neglecting second-order terms of dv/dx, we have 

S y = Ssm6= S — 


dS y = ^-\S :;, 


d 2 v d i dv" 

0*v o 0v\ 

pA o7* = dA s Tx) 

This is a nonlinear relation, as S depends also on v. However, if the initial 
tension is strong and vibration v small, S may be taken as a constant. We 
then have 

d 2 v a d 2 v 

dt 2 p dx 2 
where o = SjA. 

(b) lateral vibration of a slender bar 

When a slender bar enters into a transverse vibration, the stress and 
strain distributions produced in the bar are approximately the kind 
described by the elementary theory of beam for static bending under 
lateral loads. That is to say, the inertia forces due to such things as shear 
strain and anticlastic deformation of the cross sections can be ignored. 

Let us choose a right-handed coordinate system such that the z-axis is 
the centroidal axis of the bar and the zz-plane is the neutral plane of the 
bar during symmetrical bending. The deformed shape of the bar is then 
described by the equation of the so-called elastic curve, which lies in the 
#2/-plane (Fig. 84) : 

v = v(x, t) 

in which v is the displacement of points on the centroid axis from its 
unstressed position. For small deformation, the flexural formula states 

d 2 v 
Ox 1 




in which M is the internal bending moment and /is the moment of inertia 
of the cross section. 7 The elementary theory of bending states further 
that in the absence of externally applied moment 


-=— = V = internal shear 



= fv 



in which/^ is the distribution of externally applied force in the ^-direction. 

Figure 85 

In a dynamics problem f v is replaced by inertia force. Hence 

d 2 v dv a 2 / d 2 v 

P A 

For bars with uniform cross sections we have 

EI dh- 
pA dx* 



Let us insert a few remarks about our assumption of slenderness of the 
bar. In the deviation of (10) and (11) this assumption enters into a number 
of places. As a rough guide we may say that most of the results from 
elementary theory of bending are valid when the square of the depth-to- 
length ratio of a beam, (c/L) 2 , is much less than one. This is so in static 
bending, where the applied loads are predominantly in one direction and 

7 The sign conventions used here for bending moments and shear are those commonly 
adopted by modern textbooks on strength of materials. It is also permissible to consider 
the sign conventions as being defined by (7) and (8). 


the length of the beam enters as the distance between points of zero 
deflections or of zero moment. In vibratory motions, however, a beam 
may have deflections of the form /" "v or of the form AAAJ . Thus a 
given bar may be considered as slender in one kind of vibration but not 
in the other kind. The quantity L is therefore more appropriately inter- 
preted as the "wavelength" of the deflected form during vibration. 

If the ratio c/L is small but not altogether negligible, whereas (cjL) 2 
is negligible, the equation of motion may be corrected for the inertia 
effect of the rotation of the cross sections. The correction begins with (8). 


— — dx = V dx — moment of inertia force on dx 



dM , , a 2 (dv\ J 

^dx=Vdx + P I-y-)dx 

Together with (7) and (9), this relation leads to 

a 2 / d 2 v\ dH d*v 

It is interesting to note that this equation is not in the general form of (2). 
The reason for this deviation is that in our scheme of describing the 
deformation of the bar we think of it as being made of elemental rigid 
slices, each having a translation v and a rotation dv/dx. Hence the dis- 
placement function v is not really the displacement of a generic mass 
element in the elastic body. 

Depending on end conditions, appropriate boundary conditions must be 
prescribed. A few of the frequently encountered boundary conditions 
are as follows (Fig. 86): 

At a fixed or built-in end, 

v = 



a pinned end, where M - 

17 = 

= 0, 

d 2 V 

dx~ 2 = ° 

At a free end, where M = and V = 0, 

/ d 2 v\ 

— -0 and - 
dx 2 dx 

Still other types of boundary conditions are possible. These are discussed 
in specific problems. 




For this problem those boundary conditions involving only v and dvjdx, 
which describe the displacement and the slope at an end, are called imposed 
or geometrical; those involving d 2 v/dx 2 or d 3 v/dx?, which describe the 
moment on shear at an end, are called natural or dynamic. 

The initial conditions required to specify a particular solution of the 
vibratory motion are given by 

*>Cm ) =f( x ) 



(?, t ) = g( x ) 


3*2 ~ U 




Figure 86 


( £ '0) = ° 


These problems are two-dimensional extensions of the problems for 
vibrating strings and beams. We shall not derive the differential equations 
of motion here, since what is involved in the derivations belongs to the 
theory of elasticity. It is not difficult, however, for readers to accept the 
following results. 

For transverse vibration of a membrane lying in ^-plane when at rest 
the differential equation of motion is 

d 2 w /a 2 d*\ 


in which h is the thickness of the membrane and S is the surface tension in 
the membrane, which is assumed to be constant. For a membrane 
initially stretched to a constant surface tension the only boundary con- 
dition physically realizable is that 

w = 

on all boundary points. This equation, as well as its one-dimensional 
and three-dimensional counterparts, is called the wave equation. These 


equations are studied extensively in acoustics and in hydrodynamics. 
In other applications the boundary conditions may also contain normal 
derivative dw/dn, which is the rate of change of w along a direction normal 
to the boundary of the system. 

The equation for the vibration of a thin plate is the two-dimensional 
extension of that for a beam. It reads 

d 2 w __ / a 2 ( jp 
ph ~d? " w + df 

V 2 (E'I'V 2 w) 

i v h d 2 w\ 


in which E' — Ej(\ — v 2 ), v = Poisson's ratio, / = /z 3 /l 2, and h = plate 

4.3 Separation of Time Variable from Space Variables — 
Reduction to Eigenvalue Problems 

Most of the boundary-value and initial-value problems used for illustra- 
tions in Art. 4.2 can be solved under a number of circumstances. 
Before we get into the details of solving particular problems and thereby 
lose sight of the over-all picture, let us study our problem in general terms 
for a little while longer. Without losing too much generality, we do wish, 
however, to restrict the discussion to equations having only a single 
dependent variable, 8 which we shall call q in all cases. 

All of the equations used for illustrations in Art. 4.2 can be put into 
the general form of 

Mq+Lq = (15) 

in which M and L are two linear differential operators involving only 
space variables x, y, and z, and the dots over the q represent partial 
differentiation with respect to time. For instance, in (12) 

M - ( pA - pi S) 

d 2 

d 2 I d 2 \ 

and in (14) 

M = ph 

L = V 2 (£7'V 2 ) 

8 It is also possible to write a single vectorial equation to replace the three scalar 
equations in (2). 


In form, at least, if in nothing else, (15) resembles (13) of Chapter 3. 
This resemblance gives us ideas about possible subsequent developments. 
As we recall, what we used to do in studying free vibrations was first to see 
if the system could vibrate in such a way that all the coordinates varied as 
a harmonic function of time. To this end we let 

q.(t) = r, cos (cot - a) 

Similarly, for the present problem, we may let 

q(z, y, z, t) = r(x, y, z) cos (ojt - a) (16) 

Introducing this assumption 9 into (15), we have 

-co 2 Mr + Lr = (17) 

This is a differential equation containing no t, just as the matrix equation 
(16) in Chapter 3 does not contain t. A more formal way of arriving at (17) 
is by a method called the separation of variables. Here we let 

and (15) becomes 


Since the first ratio cannot contain t and the second cannot contain 
x, y, z, neither ratio can contain any variable and / must be a constant. 
Furthermore, it is not difficult to see that ?. must be positive so that p 
will not be an exponential function that grows with time. Thus by setting 
X = co 2 , we have 

p + co 2 p = 

Lr = co 2 Mr (19) 

These two equations share in common only the parameter co 2 , whereas 
both the dependent variables and the independent variables in them are 
different. For every solution we can find for this pair we have a solution 
for the original equation (15). But the reverse is not true, since a solution 
of (15) may or may not be expressed in the form of (16) or (18). We can, 
however, entertain hopes that since co 2 is yet undetermined there is more 
than one set of solutions for (19) and the totality of linear combinations of 
these solutions may include all the solutions of (15). This is again a 

9 For the time being the possibility of a solution in the form ( 1 6) is only an assumption 
yet to be proven. 

q = 

r&, y, 



' + pLr 




= _P 

= / 


situation similar to that encountered in Chapter 3, where a particular 
solution in free vibration is representable by a linear combination of 
principal modes. 

So far the discussion has been centered around the differential equation 
of motion which does not by itself define a physical problem. The accessory 
conditions concerning the initial values and the boundary values must also 
be considered. Let us take up the boundary conditions first, since these 
are an integral part of the problem that defines the possible ways in which a 
system may vibrate. The boundary conditions can be expressed in the 
general form 

1%) = 

in which N is a linear differential operator of the space variables and 
sometimes also of the time variable /. If N does not contain /, the replace- 
ment of q by rp according to (18) will result in 

N(r) = 

because N(q) is linear and homogeneous in q. If N contains t, it is 
because the boundary condition says something about an inertia force. 
In this case / enters only in the form of d 2 /dt 2 . Since 

if q is in the form of pr, the boundary M(q) leads to N'(r) where N' is 
obtained by replacing d 2 /dt 2 in N with -co 2 . 

To recapitulate, we now have the problem of finding a function r(x, y, z) 
that satisfies a linear homogeneous differential equation 

L(r) = co 2 M(r) 

and a set of linear homogeneous boundary conditions of the form 

N(r) = 

where co 2 is a parameter yet unspecified and N may or may not contain co 2 . 
In many vibration problems in which M is merely a multiplication 
factor that can be incorporated in L, the differential equation simplifies 

L(r) = co 2 r 

In any event, the problem is markedly similar to the eigenvalue problem 
we dealt with in Chapter 3. We expect, and we shall find, that the situation 
is as follows: 

(i) The problem has a nontrivial solution only when co 2 takes on one of 
a set of values called eigenvalues co 2 , to 2 , .... To each eigenvalue is 


associated a solution r t (x, y, z) called the eigenfunction belonging to a>?; 
together they describe the frequency and the deformation of a natural 
mode of vibration of an elastic body. 

(ii) Since the degrees of freedom of an elastic body are infinite, there 
are infinitely many natural modes. All motions of a given elastic body 
in free vibration can be represented by a linear combination of its natural 

Let us now analyze a few of the simpler problems illustrated in Art. 

(a) longitudinal vibration of a slender bar of 
uniform cross sections with a concentrated 
mass m attached to one end while being held 


The differential equation and its boundary conditions are 

E d 2 u _ d 2 u _ 

p dx 2 dt 2 
At x = 0, 

u = 
At x = L, 

d 2 u EA du „ rm ^ 

dt 2 M dx 

Let /u be the ratio of the attached mass to the mass of the bar. Then 


M = apAL — = 

ry M fiLp 

The eigenvalue problem results from letting u(x, t) = p(t)r(x): 

d 2 r co 2 p n ,_. 

* 2 + ir = (2I) 


For simplicity let 

r(0) = 


r'(L)-/*L^r(L) = (22) 

P 2 = ^ 
h E 

The solution of (21) is evidently 

r = D cos fix + B sin fix (23) 




To satisfy r(0) = 0, D must vanish and p (or oj) must be real. To satisfy 
the remaining boundary condition, we must have 


B(P cos PL - juLp 2 sin |&L) = 
pPL = cot PL 


This equation in p (or w) is the frequency or characteristic equation of the 
system. It is equivalent to the determinant equation in the matrix eigen- 
value problem. The coefficient B in (23) is indeterminate. Without losing 
generality, it may be taken as unity. 


u(x, t) 

M = [ipAL 

Figure 87 

The equation (24), being transcendental in nature, has an infinite number 
of roots. The first few positive roots 10 for four values of /u are shown in the 
following tabulation : 

h L 


LI = 






11 77/2 

t* = i 







v = 1 







Jil = 00 






The condition /u = corresponds to a free end at x = L. In this case 
Pi = (2/ — X)-n\2L. The condition ju — 00 corresponds to a fixed end at 
x = L; there ft = (/ — 1)tt/L. With the exception of these two extreme 
cases, the eigenvalues are not spaced regularly apart. But, for all ju > 0, 
Pi approaches (1 — X)tt\L rather quickly. This is understandable, since 
the inertia force goes up with the square of frequency and at high frequen- 
cies the end mass becomes nearly immobile. 

The negative roots represent nothing new when put into (23). 


Returning to the function p, we see that 

p + j 2 p = 


p = C cos (co? — a) 

= C cos [V(EI P )Pt - a] 

A typical solution to the boundary-value problem (20) is then 

u(x, t) = r(»/?(/) = C sin /3a: cos [V(Elp)pt - a] 

in which /? is any one of the solutions to the frequency equation (24). 
Since the problem is linear and homogeneous, by the application of the 
superposition theorem we obtain the following series solution to the 
boundary- value problem : 

u(x, = 2 Q sin ft* cos [V(£/p)/V - «J (25) 


The constants C and a are arbitrary as long as they produce a convergent 
series, and the function u(x, t), so defined, is at least twice differentiable 
with respect to both x and t in the ranges 11 < x < L and t > 0. With 
the freedom of choice in the C's and a's, we hope to be able to match the 
initial conditions of a given problem and to arrive at a particular solution. 
Let the initial conditions be given by 

u(x, 0) = / (x) and I — = g(x) 

If we know how to expand these two functions into two infinite series of 

u(x, 0) = f(x) = | /, sin fa 


\ot/ t=Q ? -=i 

we can determine the constants in (25) by the coefficients f t and g { . By 

putting t = in (25) and its derivative and comparing the results with 

(26), we have 

C cos a f = fi 
,— (27) 

V(E/p) ftC, sin a, = g< 

11 This differentiability requirement is a little more stringent than mathematically 


For the special cases of ju = and /u = oo the eigenvalues are spaced 
regularly apart; the two series in (26) are Fourier sine series and they can be 
obtained in the standard way. For other values of /u the series in (26) are 
no longer Fourier series in the ordinary sense of the term. The method of 
expanding given functions into infinite series of eigenfunctions is discussed 
in Art. 4.4. 

(b) lateral vibrations of bars having uniform 
cross sections 

The differential equation of motion as derived in the last article is 

-*!-£?* (28) 

To separate the time variable from the space variable, we look again for 
solutions of the type 

v(x, t) = p(t)r(x) 

The substitution of pr for v results in two ordinary equations: 

p + co 2 /? = (29) 

£-/»-0 (30) 

in which 

OA P A0)2 



The solution for (30) is 

r = A sin fix + B cos fix + C sinh fix + D cosh fix (32) 

The eigenvalue problem is now as follows: 

Given a set of end conditions, determine the set of eigenvalues 12 of 
P so that the constants A, B, C, and D will not vanish together, and find 
the ratios A \B :C :D corresponding to the eigenvalues. 

Consider a cantilever beam which is fixed at one end and free at the 
other. The boundary values are (Fig. 88) 

r(0) = r'(0) = r"{L) = and r m {L) = 

in which L is the length of the beam. From the first two of these conditions 
we have 

B + D = and A + C = (33) 

12 It is immaterial whether we call /S or co the eigenvalue, since they are definitely 
related by (31). 


From the last two conditions we have 

—A sin pL - B cos fiL + C sinh PL + D cosh 0L = 
-A cos pL + 5 sin 01 + C cosh 0L + D sinh 0L = 


C(sin PL + sinh PL) + £>(cos 0L + cosh PL) = 
C(cos pL + cosh 0L) + D(sinh pL - sin 01) = 

1 jj^W 


Figure 88 

For nontrivial solutions the determinant formed by the coefficients of C 
and D must vanish. Hence 

sin PL + sinh PL cos PL + cosh PL 
cos PL + cosh /5L sinh PL — sin /iL 


cos PL cosh PL = -1 (35) 

This is then the characteristic or frequency equation of the problem. It 
has an infinite number of solutions. The first six are 

P X L P 2 L P Z L p 4 L P 5 L p 6 L 

1.875 4.694 7.855 10.996 14.137 17.279 

To each value of P there is an co given by (31). 

The eigenfunctions are obtained by substituting (33) and (34) into (32), 

r(x) — sinh Px — sin px + y(cosh Px — cos fix) (36) 


D sinh pL + sin PL 


C cosh PL + cos pL 

We thus have a set of eigenvalues and eigenfunctions representing the 
frequencies and the shapes of the deformed bar of the natural modes. 

Mathematically, because the boundary-value problem is linear and 
homogeneous, its general solution can be represented by an infinite series 
of the eigenfunctions that is convergent in the interval < x < L. 


v(%, = 2 Q cos ( C( V — <*-t) r i( x ) (37) 


To find the particular solution under a given set of initial conditions, 

v(x, 0) = f(x) and v(x, 0) = g(x) 
we first expand f(x) and g(x) into two infinite series of eigenfunctions. 

«(*) -f *•<<*) (38) 

Upon putting / = in (37) and its derivative with respect to /, we have 

ft = Q cos a< 

gi = Qoj . sin a, (39) 

The solution for other end conditions can be obtained in the same manner. 
The reader should try his hand at verifying the following results: 

(i) Bars with hinged ends. 

r(0) = r"(0) = r (L) = and /'(L) = 

Frequency equation and eigenvalues: 

sin PL = (3L = n, 2tt, 3tt, • • • 

Eigenfunctions : 

r(x) = sin fix 

(ii) iters with free ends. 

r"(0) = r'"(0) = r"(L) = and r'"(L) = 
Frequency equation and eigenvalues: 

cos fiL cosh pL = 1 
0L = 0, 4.730, 7.853, 10.996, 14.137, 17.279 

r(x) = sinh fix + sin fix + y(cosh fix + cos /fa) 

sin /5L — sinh /3L 
cosh PL — cos /?L 

(iii) Bars with fixed or built-in ends. 

r (0) = r'(0) = r(L) = and r'(L) = 


Frequency equation and eigenvalues: Same as (ii), except that 
pL = has no physical significance. 


r(x) = sinh fa — sin fa + y(cosh fa — cos fa) 

sinh fiL — sin fiL 
cos f$L — cosh f>L 

(iv) Bars with one built-in end and one hinged end. 

r(0) = r'(0) = r(L) = and r"(L) = 

Frequency equation and eigenvalues: 

tan PL = tanh fiL 

pL = 3.927, 7.069, 10.210, 13.352, 16.493 

Eigenfunctions : 

r(x) = sinh fa — sin fa + y(cosh fa — cos /fa) 

sinh PL — sin ^L 

V = 

cos /?L — cosh /?L 


The differential equation of motion is given by (13) to be 

a* s _ 2 

= — V w 

d*t P h 

To separate the time-variable, we let 
w = pit) - r(x, y) 

= C cos (tor — <x)r(x, y) 
and obtain the partial differential equation for the eigenfunction 

VV + p 2 r = (40) 


D9 P n0 >' 


The original boundary condition w = becomes r = for (40). 





This time, after t is separated, we are still left with a partial differential 
equation containing two space-variables. Except for numerical methods, 
there is no general way of solving a partial differential equation. Only 
when the boundary conditions are simple can the solution be expressed in 
terms of familiar functions — trigonometrical functions, exponential 
functions, or Bessel functions. 13 But whether or not a solution can be 
expressed simply, it has certain analytical properties. These we like to 

r (x, y) 


I I l M 1 I 

Figure 89 

emphasize. To give an example of a solution to the boundary-value 


r = 0on the boundary 

let us consider the case of a rectangular membrane bounded by sides 
(Fig. 89). 

x = y = x = a and y = b 

For this problem we may again use the method of separation of the 
variable by trying to find a solution of the type 

r(x, y) = X(x) • Y(y) 

Following a procedure similar to that used to separate t, we obtain 

m-nx mry 

r(x, y) = sin sin — — 

a b 

13 This is understandable. After all, nature is capable of infinite varieties, whereas 
we have only a limited number of familiar functions. 



(m\ 2 ln\ 2 B 2 

in which m and n are positive integers. The eigenvalues for the problem 
are therefore all values of p that can be obtained from (42) by using 
integers for m and n. The complete solution is then the double series 

^ ^ mrrx firry 
w= 2 2 C TOW cos (co mn / - a) sin sin 

ni=l«=l a (2 

where, according to (41) and (42) 

In general, the /5's, the eigenvalues, and the co's, the natural frequencies, 
are distinct for distinct pairs of integers m and n. But if a and b are 
commeasurable, that is, if a/b is a rational number, then there are some 
repeated eigenvalues, For instance, if a\b = f , o> 34 = co 62 , oj 36 = co 92 , etc. 

4.4 Orthogonal Property of Eigenfunctions 

We have seen in Art. 4.3 that in seeking a particular solution to a free 
vibration problem it is necessary to expand a given function that describes 
an initial condition into an infinite series of eigenfunctions. For this 
expansion process to be carried out in a simple way it is desirable that the 
eigenfunctions form an orthogonal set. For instance, it can be shown that 
the eigenfunctions of a vibrating cantilever beam obtained in Art. 4.3(b) 

r irj dx = i=^j (43) 


where r(x) is defined by (36). With this established, the coefficients f t and 
g { in (38) and (39) can be obtained by the well-known procedure for the 
determination of the coefficients in a Fourier series. The principle is as 
follows: if f(x) can be expanded into 

A*) = X//X*) 

3 = 1 


rHx)f(x) = fuixy^x) 



By integrating both sides over the interval to L and utilizing (43), we have 

\ L f(x)r i {x)dx=f i \\r i {x)Tdx 

Jo Jo 

In this way/i is found. By finding g { in the same way, the coefficient C t 
and the angle oc t - in (37) can be determined through (39). 

On the other hand, the eigenfunctions in Art. 4.3(a) are not orthogonal 
in the sense of (43), except when ju = or oo. Thus the same procedure 
cannot be used without alteration to find the constants in (25) and (26). 
The treatment of this case is taken up later. 

Although it is possible to investigate the orthogonality of the eigen- 
functions for each case, it is much better to begin with a general discussion. 
For this purpose let us return to Chapter 3, where we were dealing with 
eigenvalue problems of the matrix equation : 

Cr = lAr 

We showed in Art. 3.5(b) that the eigenvectors have the property that 

?<*> C r<>> = 

when X t ^ A ; 
p«> A r<>) =0 

In the special case in which A or C is a scalar matrix we have [see Art. 


fW r (i) - o 

Here we are dealing with differential equations of the form 

L(r) = AM(r) (44) 

The corresponding orthogonal relations to be investigated are 


when X t ^ A,- 

r,M(r,) da = 


where the integration is over the domain 2 occupied by the elastic body. 
Note that the orthogonal relation for vectors involves a finite summation 
process, whereas for functions it involves an integration process. 

Let us recall that in Art. 3.5(b) the orthogonal relation for eigenvectors 
is established upon the facts that the eigenvalues are distinct and that for 
any two vectors u and v the symmetry of the matrices A and C leads to 

u (Av) = v (A u) 

u (C v) = v (C u) 


For our present problem we have a similar situation. Two eigenfunctions 
having different eigenvalues will satisfy (45) if the differential operators 
L and M, together with the boundary conditions, are such that for any two 
functions u and v satisfying the boundary conditions and certain continuity 
and differentiability requirements the following relationships hold: 

u\-(v) da = vL(u) da 
E * (46) 

j uM(v)da= j vM(u)da 

In mathematical terminology we call a boundary-value problem self- 
adjoint when the foregoing relations are satisfied. To prove that the eigen- 
functions of a self-adjoint boundary-value problem satisfy the orthogonal 
relations, as defined by (45), we need only to follow the same steps dis- 
played in Art. 3.5(b). Since eigenfunctions satisfy (44), 

lr, = A,Mr, 


r 3 Lr z = X i r j \Ar i 


Lr, = AjMrj 


r i^- r j = h r i^ r 3 


J OM - r i lr i) da = I (ViMr, 

x/jMr,) da 

Because of (46), the left-hand side is zero and the right-hand side can be 
simplified by factoring. The result is 

= (A, - X t ) I r { Mr,- da 
If X t ^ X j whenever i ^j, 

r 2 Mr, da = r,Mr ? da — 


r^-Vj da = r,Lr 2 da = 

The validity of (46) depends upon the nature of the operators L and M. 
as well as the boundary conditions of the problem. In the remaining part 
of this article we shall examine the existence of orthogonality among the 
eigenfunctions for the systems analyzed in Art. 4.3 by investigating the 
validity of (46) when applied to the differential operators in these problems. 


The differential equations studied in (a) and (b) of Art. 4.3 can be put 
into the general form of 

dx n \ dx n J 
where (47) 

m{x) — p(x)A(x) 

When n = 1, this equation governs the eigenfunction for longitudinal 
vibrations of bars, lateral vibrations of a stretched string, and torsional 
vibrations of shafts. When n = 2, it governs lateral vibrations of slender 
bars. The parameters k and m represent the stiffness distribution and the 
mass distribution, respectively, along the bar, the shaft, or the beam. In 
general, these parameters are functions of x, but frequently they are 
constants. In the latter case the equation simplifies into 

^r=(-l)"/? 2 V (48) 


mw 2 


Referring to our previous notation, we see that the two differential 
operators L and M are 

d n 

dx n 

. dx n . 

= D n [k(x)D n ] (49) 


M = m(x) 

where D stands for djdx. 

Since the operation by M is merely a multiplication by a function m(x), 
it is obvious that the second of the two equations in (46) is automatically 
satisfied. We must concentrate now only on establishing the first equation. 
Let us take the boundary of the system to be at x = and x = L. 

Casel. L = D(A:D) 

For this case (47) becomes 

dx \ dx, 
Or more simply 

— I A: — = —mcrr 
dr. \ dx] 

(kr')' = —mo 2 r 


The group of vibrational phenomena governed by this equation is discussed 
in Art. 4.2(a). The difference between the two sides of the equations in 
(46) is then 

f [ul(v) - vl(u)] dx = f [u(kv'y - v(ku') f ] dx 

Jo Jo 

Through integration by parts, we obtain 

[ul(v) - vl(u)] dx = [ukv' - vku'}% (50) 



If the boundary conditions are such that any two continuous and differen- 
tiate functions u and v satisfying them will make the right-hand side of 
(50) vanish, then the problem is self-adjoint, and its eigenfunctions are 
mutually orthogonal in the sense of (45), provided that the eigenvalues are 
distinct. In other words, if 

[r&rf) - r/fc/-/)tf = (51) 

it follows that 



f VW dx = [ L ri (x)[k(x) r ;(x)\ dx = 

Jo Jo 

I r t M(r,) dx = -mixy&y/x) dx = i^j (52) 

Jo Jo 

The second of the two conditions is the one we need for the expansion of a 
given function into an infinite series of eigenfunctions. 

Let us now examine the boundary conditions for which (51) is valid. 
It can be shown by simple substitution that if the two boundary conditions 
are both of the linear homogeneous form 

ar + br' = (53) 

then (51) is always satisfied. In (53) the coefficients a and b may have 
different values at different boundary points but they must not depend on 
the eigenvalues of the problem. Take the case of longitudinal vibrations 
of bars as an illustration. The general form (53) includes such common 
end conditions as 

(i) fixed end r = 
(ii) free end r' — 
(iii) end terminating in an elastic spring, K, 




r r 



When the condition at each end of a bar belongs to one of the three 
kinds described, the eigenfunctions are orthogonal to one another with 
respect to the weighting function m(x). 

m{x)rjx)rj(x) dx = i^j (54) 


The weighting function m(x) is the mass distribution along the bar 

m(x) = p(x)A{x) 

For homogeneous bars with uniform cross sections m is a constant, and 
we have the ordinary orthogonal relation 

\ L r i (x)r j (x)dx = i^j (55) 


Let the expansion of a given function /(#) into a series of eigenfunctions 
satisfying (54) be 

/(*) = £/w«) 

The coefficients /can be found by 

Jm(x)r i (x)f(x) dx = f t m(x)[r i (x)] 2 dx (56) 

o Jo 








Let us now return to problem of Art. 4.3(a), in which one end of the bar 
terminates in a concentrated mass M. The boundary condition at such 
an end is given by (22) as 


Even though this boundary condition is linear and homogeneous, it does 
not belong to the type described by (53) because the second coefficient 
contains o>, which is not the same for all eigenfunctions. Therefore, the 
eigenfunctions in this problem sin f$ t x, in which p { are the roots of (24), 
do not satisfy (55). In other words, 


sin ftp sin fife dx ^ 



This lack of orthogonality not only causes difficulty in the finding of the 
coefficients f t and g t in (26) but also invalidates a number of important 
analytical results based on the orthogonality or eigenfunctions. Fortu- 
nately, by interpreting the problem in a different way, the desired property 
of orthogonality can be preserved. Let us observe that in the initial 
formulation of the problem a bar of uniform mass distribution is the 
vibrating elastic body under consideration, whereas the existence of an 
attached concentrated mass M enters into the problem only as a boundary 


Figure 90 

condition. The mass distribution of the system is therefore considered 
uniform as given by 

m(x) = pA = constant 

We can, however, reformulate the problem by considering the concen- 
trated mass as a part of the vibrating bar, which will then have a free end. 
Suppose at first we redistribute the concentrated mass uniformly along a 
small portion of the bar near the right end of length e. The right end is 
then free (Fig. 90). Let the eigenfunctions for this mass distribution be 
denoted by 

Me,x) i=l,2, ••• 

in which e serves as a parameter. By physical reasoning, 

lim 0,-(€, x) = sin p. ( x 

since a concentrated mass is the limiting case of a finite mass distributed 
over an infinitesimal region. In the meantime the eigenfunctions are 
orthogonal with respect to the modified mass distribution function mfc, x) 


w(€, xyt>l€, *)<£/€, x) dx = i+j (58) 

m(e, x) = pA < x < L — e 
m(e, x) == pA + jupAL/e (L — e) < x < L 


Because of (58), we can apply (57) and take the limit as e ->0. Consider 
the numerator of (57) first. 

m(e, #)</>;(€, x)f(x) dx 



- P A 

^ V/€, X)/(X) dx+(\+ ^U _ &(*, *)/(*) <fc 

For sufficiently small e the integrand in the second integral must be 
continuous. Hence we may use mean-value theorem to obtain 

Jm(e, ^)</> z (e, x)f(x) dx 

= p a\ J V*fo *)/(*) * + (€■+ ^W*. f)/tt) 

where L — e < £ < L. 
Hence 14 



m(e, x)^^, x) f (x) dx 


= p/4 J sin PfX f(x) dx + //L sin p t Lf(L) 


lim m(c, #)[<^(e, x)] 2 dx 


= p^ I sin 2 &# cte + juL sin 2 /3,-L 1 

sin pixfix) dx + //L sin ^Lf(L) 


The result is 

J sin 2 j8 t -x afc + /uL sin 2 &L 

Although the formula just derived is for a concentrated mass at one end of 
a uniform bar, the principle behind the derivation is applicable whenever 
concentrated masses produce eigenfunctions not orthogonal in the ordi- 
nary sense. Mathematically speaking, the mass distribution function m(x) 
in the general formulation of the problem described by (2) is allowed to 
behave like Dirac's ^-function, with singularities at a finite number of 
points, yet bounded elsewhere and integrable over the domain occupied 

14 We are using a convergence theorem, attributed to Lesbeque, which permits the 
passing of the limit sign from the outside to the inside of the integral. 


by the system. When the existence of concentrated masses is treated in 
this way the eigenfunctions are again mutually orthogonal with respect to 
this modified density function. 

Another broad implication of the example studied is that, although in a 
differential equation formulation of a dynamics problem we may be forced 
to define the boundary of our systems in such a way that no concentrated 
forces may appear at the interior points in formulations by integral 
equations or energy equations, such restrictions may not be needed. This 
aspect of the problem is to be amplified later. 

Case 2. L= D 2 (kD 2 ) 

Having studied the case of n = 1 in (49), we can now follow the same 
procedure for n = 2. This case pertains to the lateral vibration of bars or 
beams. The eigenvalue problem is embodied in the differential equation 


d 2 


(kr")" = moj 2 r 

where k(x) = EI = bending stiffness of cross sections 
m(x) = pA = mass distribution along the bar 

The parameters E, /, A, and p may be constants or variables. The next 
thing is to investigate the conditions under which the first equation of (46) 
is valid. The second one is obviously valid. Again, through integration 
by parts, we have 

Pi/L(i?) dx = \ L u(kv"y dx = [u(kv")']% - f L u '(kv")' dx 

J Jo Jo 

= [u(kv")' - u\kv'X + f u'kv" dx (59) 


I vl(u) dx = [v(ku"Y - v\ku")]L + I v"ku" dx 

Jo Jo 

Since the integrals at the right-hand sides of the two equations are the 
same, the difference between the left-hand sides is 

[ul(v) - vl(u)] dx = [u(kv'J - v(ku"Y - u'kv" + v'ku'% (60) 

Again the boundary conditions of the problem will determine whether it is 
self-adjoint and its eigenfunctions, mutually orthogonal. The physical 
meaning of the terms at the right-hand side of (60) should now be reviewed. 


It is seen that 

u, v = deflections at the ends 
u', v — slopes at the ends 
ku ', kv" = bending moments at the ends 
(ku")' ', (kv") = shear forces at the ends 

Therefore, for any of the following simple end conditions each of the four 
terms at the right-hand side of (60) vanishes individually. (See Fig. 86.) 

At built-in ends r = r = 

At pinned ends r = kr" = (61) 

At free ends kr" = {kr")' = 

A more general way whereby (60) will vanish is that each end condition 
is a linear homogeneous relationship either between the deflection and the 
shear or between the slope and moment. Analytically, it means that at 
each end there is a set of constants a, b, c, and d such that the boundary 
conditions may be expressed by 

ar + b(kr"Y = 
and (62) 

cr' + d(kr") = 

The conditions stated by (60 are special cases of (62). The reader can 
verify that if u and v satisfy (62), then (60) vanishes. Physically, in the 
most general case (62) represents the end conditions when the ends are 
elastically supported. The stiffness of the support may be zero or infinite. 
Sometimes a bar terminates in a concentrated mass. At such an end 
the linear relationship between the deflection and shear depends upon 
frequency. The orthogonal relation among the eigenfunctions will then 
have to be modified in the manner already discussed. The result is 



m(x)r i (x)r j (x) dx + Mr^Qr^L) = i ^ j 

where M is the concentrated mass at x = L. 

Case 3. L = V 2 

The two-dimensional wave equation (13), after separation of time 
variable f, reduces into 

V 2 r = —o 2 kr 

k(x, y) = phjS 


The integral to be examined this time is 


(u V 2 v - v V 2 w) dA (63) 

This integral is over a two-dimensional region A. Our previous technique of 
integration by parts cannot be used here, but there is a procedure that is 
entirely equivalent. This procedure is best described with notations in 
vector analysis. It is known that the Laplacian operator V 2 is the diver- 
gence of a gradient 

V 2 = div (grad) = V • V 

u ^v = wV • (Vv) = V • [u(Vv)] - (Vw) • (Vr) 

v V 2 w = V • [v(Vu)] - (Vr) • (Vt/) 

Substituting into (63), we have 

f (u V 2 v - v V 2 w) dA= f V • [w(Vr) - v(Vu)] dA 

Now we apply divergence theorem to the right-hand side and obtain the 
statement of Green's theorem. 


(u V 2 v -r v V 2 w) dA = (b [u(Vv) - r(Vw)] ■ ds 

The right-hand side is a contour integral around C, which is the boundary 
of A. It can also be written as 

I (u V 2 v - v V 2 w) dA = <j> (u^- - v^\ ds (64) 

in which djdn stands for "the normal derivative of" at the boundary. 
Hence, if the boundary condition of the problem is either 

r = on C 


^- = 0onC 

then (64) vanishes and the eigenfunctions with distinct eigenvalues are 
orthogonal in the sense that 


k(%, y)ri( x , yVjix, y) dx dy = (65) 

(O t ^ COj 




The special case of two different eigenfunctions with the same eigen- 
value can be handled by a principle similar to that used in Art. 3.10. 
If r t and r j are two independent eigenfunctions with the same eigenvalue 
and they are not orthogonal to each other in the sense of (65), it is possible 
to replace one of the two functions by a certain linear combination of the 
two functions, which is also an eigenfunction and satisfies (65). We shall 
not go into the details of finding this combination. After all, the situation 
is highly pathological because a very small change in the system will 
separate the eigenvalues and result in two orthogonal eigenfunctions. 

Case 4. L = V 2 (&V 2 ) 

Having gone through three similar cases and a general discussion, we 
shall simply give a key derivation for this case. The reader can easily 
fill in the rest of the details if he wishes. 

By formulas in vector analysis 


uV 2 (k V 2 v) = V • [uV(k V 2 r)] - Vw • V(k V 2 r) 
Vw • V(k V 2 r) = V • [k V 2 v(Vu)] - k V 2 r V 2 « 

f [uV 2 (k Vh) - vV 2 (k V 2 w)] dA 
= ( V • [uV{k V 2 v) - k V 2 r(Vw) - vV(k V 2 u) + k V 2 w(Vr)] dA 

u — (k V 2 iO - k V 2 r-^ -v — (k V 2 w) + k V 2 u ~ 
on on on on 


The problem is self-adjoint if this integral around the boundary vanishes. 

4.5 Formulation by Integral Equation 

In Chapter 3 we showed that equations of motion can be written with 
either the elastic constants or the influence coefficients for positive- 
definite systems. The matrix operators formed by the influence coefficient 
is the inverse of that formed by the elastic constants, and vice versa. 
Hence from 

C r = oj 2 A r 

we can obtain the equations by influence coefficients with an inverse 

C -i Ar 





We expect that similar inverse operations may exist for the differential 
operators being studied. That is, for 




there is an equivalent relation written as 

L- 1 Mr = -I - r 

Since L is a differential operator, L _1 must denote an operation involving 
an integration process. To find the nature of this inverse operation, we 
have chosen to approach it on the basis of the physics of the problem 
rather than to discuss it as a purely mathematical theorem. (See Exercise 

Unit force 

Figure 91 

Consider for example the lateral vibration of a bar supported in some 
way. (Fig. 91). From the way the bar is supported (boundary conditions) 
and from its dimensions and make-up (operator L) we can find the in- 
fluence coefficient between any pair of interior points in the bar. The 
influence coefficient for a given system is therefore a function of two 
points, which in the case of a slender bar are designated by two variables, 
say, the distances of the two points from the left end, x and |. In mathe- 
matics this function is called Green s function and is denoted by the symbol 
G{x, £). For linear systems Maxwell's reciprocal theorem states that 

G(x, |) = Gtf, x) 

or that Green's function for a linear system is symmetrical. 

Let us now assume that the bar is vibrating in one of its natural modes. 
The displacement along the bar is then described by 

v(x, t) = Cr(x) cos ((ot — a) 


The inertia forces acting on the bar have the distribution 

d 2 v 
— m(x) — — = Cco 2 r(x) cos (cot — a) • m(x) (67) 



m(x) = p(x)A(x) 

The relation between the inertia forces on the bar and the displacement at a 
generic section x is then 

v(x, t) 



G(x,£)m(£) — v( 

£, t) d£ 

Or, after elimination of 

cos (cot 

— a), 


G(x, £); 

wtfMO di = - 2 




this equat 

ion can 

be expressed by 
L (Mr) = i- 2 r 

CO 1 



where M = m, and L _1 represents an integration operation. 

The integral equation (68) is called a linear homogeneous integral 
equation of the second kind. It states a typical eigenvalue problem : "Given 
an operation characterized by G and m, find a function r which remains 
unchanged by this operation except for a multiplication factor." Stated 
in another way the problem is to find the values of co 2 for which (68) can 
have nontrivial solution, the trivial solution being 

r(x) = 

Let us note the fact that in this formulation no boundary conditions are 
mentioned. This is because the information concerning boundary con- 
ditions is already included in Green's function of the problem G. It is a 
simple matter to show that if G(x, £) satisfies a linear homogeneous 
boundary condition 

G + aG' + bG" • • • = (70) 

in which the prime denotes differentiation with respect to x, then any 
function r satisfying (68) must also satisfy the same boundary condition. 
It should also be borne in mind that unless the boundary condition is 
linear homogeneous the superposition principle on which the formulation 
is based will no longer be valid. 


Although the lateral vibration of bars was chosen initially for this 
discussion, the formulation presented is quite general. The integral 
equation (68) represents all linear eigenvalue problems having one space 
variable. For multidimensional problems (68) is generalized into 

^G(P,Q)r(Q)da Q = ^- 2 r(P) (71) 


in which P and Q represent the coordinates of two generic points in the 
elastic body. If the displacement r is not unidirectional, we shall have 
more than one equation of the form (71). 

The usual analytical method of solving an integral equation, such as 
(68), is to find the associated differential equation, such as (66), and solve 
it. In this way, there is no advantage in formulating a problem by integral 
equation as long as we can always formulate the same physical problems 
by differential equations. However, the integral equation formulation has 
great heuristical value in numerical analysis. For instance, the form 

L- 1 Mr = — /• 

immediately suggests an iteration procedure for the determination of the 
eigenfunction with the lowest co 2 . The scheme will be the same as that 
described in Art. 3.11(b). The numerical method for solving this integral 
equation is discussed in Art. 4.11. 

4.6 Rayleigh's Quotient and Its Stationary Values 

(a) rayleigh's quotient for free vibration of an elastic body 15 

In Chapter 3 we showed that the eigenvalue problem of the matrix 

C r = ( o 2 A r 

is related to the finding of stationary values of the associated Rayleigh's 

u C u 

Q u = — - 
u A u 

Briefly we found that 

(0 Q( u ) is greater or equal to the lowest eigenvalue. Or 

<2(u) > co 2 if co 2 < co 2 \ • ■ • , < co* 

15 To gain a better understanding of this discussion the reader is advised to review 
Arts. 2.6 and 3.6. 


(ii) If u is an eigenvector, Q(u) is the associated eigenvalue. 

e(<" (i ») = co, 2 

(iii) If u is only slightly different from a modal vector, then Rayleigh's 
quotient of u differs from the corresponding eigenvalue by a second-order 
small amount. In other words, Rayleigh's quotient achieves a stationary 
value when u is one of the modal vectors. 

It is also important to remember that one essential premise for these 
conclusions is that 

u A u > u C u > 

The analysis in Art. 3.6 suggests heuristically that for the eigenvalue 
problem in this chapter 

L(r) = w 2 M(r) 

We may form Rayleigh's quotient of a function u(x, y, z) 

uL(u) da 

Q(fu)) = -^ (72) 

u M(w) da 

which perhaps will have similar properties. 

This time the value of the quotient Q depends on the form of a function 
u instead of a vector u, hence it is a "function of a function" or afunctional, 
denoted here by a double parenthesis as used in (72). 

Having noted the similarity we shall now mention the obvious difference 
between the two cases. In forming Q(u), the vector u is completely arbi- 
trary, whereas the function u in Q((u)) is subjected to some restriction. 
In the first place, the function u must be sufficiently regular in its continuity 
and differentiability so that L(w) and M(w) can be defined and the two 
integrals in (72) have meanings. In the second place, since the boundary 
conditions of the problem as well as the differential equation describe 
the physical nature of the vibrating system, these conditions must somehow 
enter into consideration. 

Let us consider a class of functions called admissible functions. 1 * which 
satisfy all the boundary conditions of an eigenvalue problem and for which 
L(w) and M(w) are continuous functions inside 2. Obviously, the eigen- 
functions themselves must belong to this class. Any function u belonging 
to this class can be expanded into an infinite series of eigenfunctions, 

16 There is no universally accepted meaning for this term. It is thus necessary to 
examine the conditions of admissibility whenever one sees the term used. 




which can be differentiated termwise to yield series representing L(V) and 
M(w); and all of the series thus obtained are uniformly convergent. 
Accepting this statement without proof, we may write 

u = v x r x + v 2 r 2 -\ = 2 v t i 


i = l 

where the coefficients v i for self-adjoint problems are evaluated by methods 
in Art. 4.4. By termwise differentiation, we have 

L(") = I>M = 5>,VM(/v) 

1 = 1 

= 1 



") = m(|^) =|^M(.,) 



Let us assume that the boundary conditions are such that the problem 
is self-adjoint. The eigenfunctions are then mutually orthogonal with 
respect to both L and M. This orthogonality, together with the uniform 
convergence of (73) and (74), enables us to write 

f* co co /• 

wL(w) (/ff = n ViVjCofcMir,) da 

i = l JS 

Similarly, 17 

I uM(u) da = f v? I ^M(r,) tfo- 

For simplicity let us write 

m it ^jr t M(r t )da (76) 

Rayleigh's quotient (72) for an admissible function u can thus be written 

G(M) = 

Wu«iV + m 22 v 2 2 co 2 2 + 

W + ^22^2 + * ' 


2 m,^, 2 co, 2 


2 "»«»<* 


17 If (74) is uniformly convergent, so (75) will be. 


Except for its infinite character, this expression resembles (39) of Art. 3.6 
in every other way. Inasmuch as both the numerator and the denominator 
are convergent series, we can use the same reasoning employed in Art. 3.6 
and conclude the following: 

(i) Rayleigh's quotient for any admissible function must be at least 
equal to the lowest eigenvalue of the system. (This conclusion is contin- 
gent upon the fact that m H is nonnegative, a fact which will become clear 
later. See also Exercise 3.11.) To show this, we need only to observe that 
(77) can be written 

__ 2 , m 22 v 2 \w 2 2 - «)*) + m 3S v s 2 (oj 3 2 - to 2 ) • • • 

Q((u)) = o h * + 

2 m u v i 2 

i = l 

(ii) If u = r it from the orthogonal property of eigenfunctions 

v ± = v 2 = • • • = v^ = v i+1 = • • • = 

fi(W) = < 

(iii) In the "neighborhood" of an eigenfunction Rayleigh's quotient is 
stationary. That is to say, if an admissible function is only slightly different 
from an eigenfunction, its Rayleigh's quotient differs from an eigenvalue by 
a "higher-order" small amount. Although this statement is not mathe- 
matically precise, it is intuitively useful. To justify this statement, we 
observe that since every admissible function u is supposed to have an 
infinite-series expansion consisting of eigenfunctions whose coefficients 
determine collectively the original function we may consider Q{{uj) as a 
function of these coefficients as given by (76): 

Q((u)) = Q(v v v 2 , • • • , r, • • • ) 

A small change in the function u corresponds to a set of small changes in 
the coefficients i\, v 2 , ■ • • , v i9 .... The corresponding change in Q, 
denoted by the symbol 6Q and called the variation of Q, is given by 

dQ = — di\ + 3— 6v 2 + • • • 

di\ dv 2 

= 1 ^— d»i 

i = l OVi 

Accepting the fact that here we can treat a function of (countable) infinite 
number of variables by the usual rules of calculus, we may easily show 


— — = when u = r ; 


for all i and/ By u = r j9 we mean that 

Vl = v 2 = • • • = y^ = p m = • • • = 

In other words 

(3Q = 

when u is one of the eigenfunctions, say r jm 

(b) minimum characterization of eigenvalues 

A better and more precise way of characterizing the stationary property 
of Rayleigh's quotient is as follows. Arrange the eigenfunctions in a 
sequence with ascending eigenvalues: 

h, r 2 , • • • , /•„ • • • 

oh 2 < oj 2 2 <,' — , co?, • • • 

Suppose that we remove from the class of admissible functions all those 
which are not orthogonal to the first (k — 1) eigenfunctions of the system. 
In other words, we select from the original class of admissible functions a 
subclass in which all the functions are orthogonal to r v r 2 , . . . , r^. In 
this smaller class the function that has the minimum Q is r k . Rayleigh's 
quotient of any other functions in this subclass will be greater than m k 2 . 
The reader is left with the proof of this statement. In proving it one needs 
only (77) and the orthogonal relation of eigenfunctions. 



</> l5 25 • • • , (/> k _ 1 

be a set of (k — 1) functions, which are completely arbitrary except that 
they are integrable in S. We remove from the class of admissible functions 
all those which do not satisfy the relation 18 

J </> ; M(w) da = j = 1, 2, • • • , k - 1 


In other words, we retain only those admissible functions satisfying (78). 
Again we end up with a smaller subclass. This subclass will be the same 
as that in (b) if the </>'s are the first (k — 1) eigenfunctions; otherwise they 

18 This relation will mean orthogonality between j> } and // if the operator M is such that 
(46) is satisfied. 


will be different. We now have the following theorem: the minimum 
value of Q formed by functions in this subclass is not greater than the 
kth eigenvalue, or 

[fi((«))Li„ < ^ 

in which u is an admissible function satisfying (78). To prove this theorem, 
we show first that there is a function u belonging to this subclass, which 
can be represented by a linear combination of the first k eigenfunctions: 

"o = *Vi + v*r 2 , • • • , v k r k (79) 

where v l9 v 2 , . . . , v k are the coefficients. If there is a w , then according 

to (77) 

n(( xx _ "%g>iV + m 22 a) 2 2 v 2 2 + • • • + m kk oj k 2 v k 2 2 

m lx v 2 + m 22 v 2 2 + ' ' • + m kk v 2 

To prove that indeed there is such a function we must show that it is 
possible to find v lf v 2 , . . . , v k for (79) so that (78) will be valid. It is not 
difficult to show that by substituting (79) into (78) we can obtain (k — 1) 
linear homogeneous equations with k unknowns, namely, v l9 v 2 , . . . , v k . 


fllVl +/l2^2 + ' ' ' +flk V k = ° 
fttPl +/ 2 2^2 + • • ' +/ 2 fc^ = ° 

f(k -1)1 V 1 + J(k-l)2 V 2' ' ' if{k-l)k V k = ^ 


f H = J 1 ^Mir^da (81) 

j= l,2,---,(/c- 1) 
i = 1,2, • • • ,k 

This set of equations (80), having more unknowns than equations, 
evidently has a solution, though not a unique solution. This completes the 

4.7 Rayleigh-Ritz Method 

(a) mathematical principle 

The discussion in Art. 4.6 reveals that if we have a way of finding in a 
class of functions the one that gives the least value for Q we shall either 
have the solution to the eigenvalue problem or establish the lower or 


upper bound of the eigenvalues. To perform this minimization process to 
a class of all admissible functions is equivalent to solving the original 
problem in differential equation, which cannot always be carried out. In 
the Rayleigh-Ritz Method we apply the minimization process to a smaller 
class of admissible functions and thereby obtain an approximate solution 
to the problem. 

We begin by selecting a set of n linearly independent admissible functions : 

Except for the requirement of linear independence and the condition of 
admissibility previously defined, the selection of these functions can be 
done in an arbitrary manner. From this set of </>'s and by taking linear 
combinations of (/>'s, we can form a class of functions, which have the 
general form 

u = ^ + q 2 cf> 2 , • • • + q n (f> n (82) 

in which the g's are a set of arbitrary coefficients. It is obvious that every 
function belonging to this class is an admissible function but not all 
admissible functions are in this class. The set of functions </> is called a 
generating set. It generates a class of functions u defined by (82). 
Let us define 

■i da 

D, da 

For self-adjoint problems 

a u = a it and c u = c H 

Rayleigh's quotient for any function u representable by (82) can be 
obtained by substituting (82) and (83) into (72). The result is 

it n 

2 2 c a<ii<ii 

Q((u)) = t=li=l 

2 2 a a a i a J 

Or, by the summation convention used in Chapter 3, we have simply 

CjMl _ qCq 


In this expression a tj and c ij are predetermined by the initial choice of 
<f>, the generating set. The q's, however, are still free to vary. The amount 
of freedom is thus equivalent to that of «-degree-freedom systems. To 
minimize Q, with respect to all possible combinations of q's, we set 

dQ dQ dQ 

dq 1 dq 2 dq n 

and Q = & (86) 

The procedure for combining (84), (85), and (86) is exactly the same as that 
leading from (26) to (31) in Art. 2.6. The result is the familiar equation 

K,co 2 - c i} \ = (87) 

In general, this equation will have n roots: d^ 2 , co 2 2 , . . . , co n 2 , arranged 
in the ascending order of magnitude. To each of these roots there associates 
a modal vector (q l9 q 2 , . . . , q n ), which in turn yields a function r through 
(82): '««,«, 

r = Wl + ?2<?2 + * * ' + qj>n 

In total we have n such functions, r l9 r 2 , . . . , r n . 

The essence of this method is that by limiting the admissible functions 
to a class representable by (82), we are approximating a system having an 
infinite number of degrees of freedom by one having n degrees of freedom. 
The n frequencies and modal vectors of the latter are then considered as 
giving an approximate description of the first n modes of vibration of the 
original system. To amplify this statement, let us observe that r x is the 
function with the smallest Q in the class of functions represented by (82). 
If this class is "sufficiently representative" of the original admissible class, 
r ± is an approximation of r l9 which is characterized by having the smallest 
Q of all admissible functions. In the meantime cb^ 2 is a better approxi- 
mation to o)^- than /\ is to r x because of the stationary property of Q. In 
any event, r 1 and w x 2 are the best approximations available within the free- 
dom of choice allowed by (82). 

There is, of course, no guarantee that the approximation will be good. 
For instance, it can conceivably happen that all of the <£'s in the generating 
set will be orthogonal to r v In that case, r x must also be orthogonal to r x 
and can never be considered as an approximation. In practice, this does 
not happen because we generally have some idea of what r x should be like 
and choose the generating set accordingly. Now in the approximating 
system it can be verified readily that r 2 is orthogonal to /\ with respect 
to M. 


Mr 2 da = 


Hence r 2 is the function with the smallest Rayleigh's quotient among all 
functions orthogonal to f x and representable by (82). This description 
parallels the minimum characterization of r 2 , except that there we consider 
a larger class of admissible functions. It is clear then that, if ^ were 
exactly r l9 co 2 and r 2 would be only approximations for oj 2 and r 2 , respec- 
tively. Since r x is only an approximation for r l9 oj 2 and f 2 in general can be 
only rougher approximations of oj 2 and r 2 . Similar things can be said for 
the rest of eigenvalue-eigenfunction pairs. 

Because we use a smaller class of admissible functions in the Rayleigh- 
Ritz method we know immediately that 

to 2 = R((rj) < RdrJ) = o\ 2 
It is also true that 

co 9 2 < tb 9 2 , co J 1 < co A • • • co 2 < m. 

but the proof of these inequalities is not within our reach at the moment. 
To avoid excessive digression we shall put the matter aside and return to 
it later in Art. 4.8(b). 

(b) liberalization of requirements for admissibility by 
energy consideration — an example 

Up till now an admissible function u eligible for consideration in the 
minimization process had to satisfy all the boundary conditions of the 
problem. Besides, L(w) and M(w) had to be defined and to be continuous 
everywhere. Both of these requirements can be liberalized. For instance, 
the continuity requirement for L(w) and M(w) can certainly be relaxed 
somewhat. In other words, to be valid, we can have a larger class of 
admissible functions and still have all the conclusions arrived at in Art. 4.6. 
To carry out this liberalization by a general and strictly mathematical 
approach is neither feasible here nor conducive to a clear understanding of 
the physical problem. Therefore, this time we choose to develop the 
theory by the use of an example. 

Consider the problem of lateral vibrations of a bar with a built-in left 
end and a right end fastened to an elastic support (Fig. 92). The separation 
of time-variable t from the partial differential equation (10) results in 

{kr") n = co 2 mr (88) 


k(x) = El and m(x) = pA 

The boundary conditions to be satisfied by r are 

r (0) = 0, r'(0) = (89) 

k(L)r"(L) = -K/iL) and [k(L)r"(L)]' = A>(I) (90) 




in which K x and K 2 represent the rotational and the lateral stiffness of the 
right-end support. Note the proper signs for the right-hand side of (86) 
in order for K x and K 2 to be positive. 

The problem is self-adjoint, since the boundary conditions are in the 
form of (62). 

When k and m are constants, this eigenvalue problem can be solved 
without much difficulty. But, if there are variations in the stiffness and 
mass distribution along the bar, the problem is usually solved only by an 



Figure 92 

approximate method based on the Rayleigh-Ritz principle. According to 
our previous discussion, we shall have to choose a set of admissible 
functions <j> x , </> 2 , . . . , <f> n9 which satisfy the boundary conditions (89) and 
(90). Although it is not too difficult to find a set of these functions, they 
may make the subsequent computation algebraically complicated. Hence 
the relaxation of boundary conditions is of practical interest. 

Let us suppose for the time being that we are still dealing with functions 
satisfying our original requirements of admissibility. The numerator of 
Rayleigh's quotient for an admissible function u is 

f uL(u) da = f u(x)[k{x)u"(x)Y dx 

According to (59), integration by parts yields 

J ul(u) da = f k{uf dx + [u(ku"Y - u'(ku")]% (91) 

Since u satisfies (89) and (90), we have finally 

I ul(u) da=[ k{u"f dx + K 2 [u{L)f + K^u'iL)] 2 (92) 

Jl Jo 


As we examine the three terms at the right-hand side of (92), we see that, 
except for a factor of J, the first term represents the strain energy in the bar 
and the last two terms represent the strain energy in the two supporting 
springs. Hence 


ul(u) da = 2V m (93) 

if u satisfies all the boundary conditions. The physical meaning of the 
symbol V m is the maximum potential energy the system would have if its 
vibration were described by 

v(x, t) = u{x) cos (cot — a) (94) 

As the denominator of Rayleigh's quotient of u, we have 

uM(u) da = m{x)[u(x)f dx = — T m (95) 

Js Jo or 

in which T m is the maximum kinetic energy the system would have if its 

vibration were described by (94). Therefore, Rayleigh's quotient can be 

expressed as 

co 2 V m 
Q = -=r* (96) 

The quantity co 2 actually does not appear because it is also contained in 
T m and will be canceled out when the energy expressions are written 
explicitly. It was also shown in Art. 2.6 that the minimization of (96) is 
equivalent to the minimization of the expression (T m — V m ); or 6Q = 

W m - VJ = (97) 

Thus we can characterize the natural frequencies of a vibrating system as 
the stationary values of the ratio in (96). This is true regardless of the 
number of degrees of freedom the system may have. 

Now we must realize that in deriving (96) from (91) we were actually 
reasoning backward. Without going into topics in advanced dynamics, 
we somehow feel that (96) is a more basic expression of Rayleigh's quotient 
than (72), since, unlike (72), it contains well-defined physical quantities 
and its definition does not contain explicitly such mathematical operations 
as differentiation and integration. Accepting as a fundamental physical 
fact that 19 

• -* m 

19 Note also that the orthogonal property of eigenfunctions needed to arrive at (77) 
also becomes unnecessary if (97) is accepted as a basic physical law. 


with T m and V m as the maximum energies in a natural mode of vibration, 
we realize that many of the requirements are imposed on the admissible 
functions simply to make the left-hand sides of (93) and (95) become the 
energy expressions. For instance, the function u must satisfy the boundary 
condition (90), so that (93) will then be valid. But, on the other hand, we 
can obtain directly through formulas in strength of material 

2V m = k(u"f dx + KMP? + K 2 [u(L)f (98) 


without the benefit of (92). 

Looking at (98), we see now that the admissibility requirements for the 
present problem are simplified : 

(i) An admissible function u must be at least twice differentiate and 
u" must be integrable between the limits and L. This is in contrast to the 
four-time differentiability needed in defining L(«), although this relaxation 
of requirement is of little practical significance, since in picking a set of 
</>'s for the Rayleigh-Ritz method we generally choose functions with 
regular behavior. 

(ii) An admissible function u for the problem on hand must satisfy the 
two boundary conditions in (89), but not necessarily those in (90), if 
Rayleigh's quotient is defined by (96) instead of (72). The difference in the 
two types of boundary conditions lies with the fact that the use of potential 
energy expressions automatically takes care of the conditions at the right- 
end elastic support, whereas they cannot take care of the fixed support at 
the left end. This is because we can include the two springs in our vibrating 
system. We cannot do the same for the fixed support without introducing 
ambiguity in the energy expression, for whenever a boundary point is 
immovably constrained the presence of an infinitely rigid and infinitely 
large external body is implied. If this body is considered part of the 
system, the energy expressions become indeterminate. If this body is 
considered external to the system, the boundary conditions it imposes must 
be obeyed by the functions eligible for consideration in minimizing (96). 
Generally speaking, if 

dx n \ dx nX 7 

all boundary conditions containing no derivatives higher than the (n — \)th 
order must always be satisfied. Such boundary conditions are thus called 
imposed boundary conditions. They usually describe constraints on the 
system that are purely geometrical; hence they are also called geometrical 




boundary conditions. On the other hand, boundary conditions contain- 
ing 20 derivatives of the nth order or higher are called natural or dynamical. 
These boundary conditions do not have to be satisfied by the admissible 
functions in minimizing (96), since they describe dynamical conditions at 
the boundary that will automatically be satisfied by the function obtained 
from the minimization process. Functions satisfying all of the imposed 
boundary conditions and having the differentiability described in (i) are 
called essentially admissible functions. 21 

Returning to our example of a bar with a built-in end and an elasticallv 
supported end, we see that the following functions are examples of essen- 
tially admissible functions : 

<p n = 1 - cos 




n = 1 



(j> n — sinh 2 



To apply the Rayleigh-Ritz method, let us choose as the generating set the 
following four functions: 

(7TX\ I 2)7TX\ I 5lTX\ 

1-cos-j ('-cos — ) (1-cos — ) and 

1 — COS 


The system is thus approximated by one having four degrees of freedom. 
The functions generated by this set are representable bv 


1 \ 7TX 


- cos in --] 

U =M n -2)L q " Sm [ n -2k 

U "= X{ n ' l 2J Z" 2 ^ COS ( ,7 -^T 

For the purpose of illustration let us assume 

k = constant m = constant 

20 It is assumed that these high-order derivatives cannot be eliminated in the set of 
boundary conditions of the problem. 

21 L. Collatz, a German author who writes a great deal on the subject, calls such 
functions simply zulassige (admissible) functions and uses the term Vergleichsfimktionen 
(compatible functions) for those satisfying all the boundary conditions. He also uses 
the adjectives wesentliche (essential) and restliche (remaining) for imposed and natural 
boundary conditions, respectively. 


The strain energy due to the bending of the bar is thus 
The strain energy in the two supporting springs is 

= 2 K i(ii + ^ + % + ? 4 ) 2 + 2 M ^) ^ ~ 3 ^ 2 + 5 ^ 3 ~ 7 ^) 2 

By putting the two together, we have a strain energy expression that is a 
quadratic form in g's: 

in which the repeated indices i and / are summed over the range 1 to 4. 
Similarly, the maximum kinetic energy expression T m is 


in which d tj is the Kronecker delta (d {j = 1, when i =j; d tj = 0, when 


With a u and c tj computed, the problem can then be solved by methods 
in Chapter 3. We solve a problem of similar nature in Art. 4.8. 

4.8 Formulation of Problem by Infinite-Series Expansions 

of Energy Expressions — Rayleigh-Ritz Method Re-examined 


In Art. 4.1 we mentioned the alternate method of specifying the con- 
figuration of a vibrating elastic body by infinite series. We now show that 
this method leads directly to an approximate solution of the free vibration 
problem, which resembles the Rayleigh-Ritz method in detail. 

Consider first the simple case of longitudinal vibrations of a slender 
bar of length L. The motion of the system is described by the displace- 
ment function u(x, t), which is the unknown to be determined. At any 

T m = \mco 2 u 2 dx 


= i^ViM? 

a i} = m 



1 _ cos(2 /-l)_ 

1 — cos (2/ — 



77 > 

tt (-1)' (-iy 

2 2/ - 1 2/ - 1 



instant t the shape (or the configuration) of the bar is described by a 
function of x. As a function of x for a fixed t, u is known to be (i) con- 
tinuous for < x < L, since the bar would otherwise be broken, and (ii) 
E(dujdx) = a x exists and is continuous in < x < L, except possibly 
at a few points along the bar where concentrated external forces are 
acting on it. 

It is known that all functions having the regular behavior described above 
can be expanded into an infinite series formed by a set of "suitably 
chosen" functions of x: l5 </> 2 , ...,</>.,... . 

u = ^ + q 2 cf> 2 , • • • , q^ • • • 

so that the series converges uniformly in < x < L and is termwise 
differentiate, except possibly at a few points. The same set of functions <z> 
can be used for different values of t. As u varies with t, the ^'s vary also. 


i = l 

The unknown of the problem now becomes a set of functions of t: q v 
q 2 , . . . , q it . . . which now becomes the generalized coordinates of the 
system, since by specifying their values one specifies uniquely the con- 
figuration of the system. 

To study what constitutes a suitable set of </>'s, we must consider among 
other things the boundary conditions of the bar. Let us assume that the 
bar is held fixed at the left end where x = 0. Hence 

W (0, t) = ^(r)^i(O) + q 2 (t)U0) • • • = 

for all /. If the ^'s, the generalized coordinates, are supposed to be 
linearly independent, it becomes necessary that 

</>,(()) = (100) 

for all /'. 

We know that a continuous function in < x < L can be expanded into 
a Fourier series with the fundamental interval — 2L < x < 2L. We choose 
a fundamental interval of expansion longer than the length of the bar L 
so that we will have more selections of the functions to be used. 22 To 
satisfy (100), only the sine functions are retained. 

sin — — n = 1, 2, 3, • • • 

22 Also, by making x = and x = L inside the fundamental interval of Fourier- 
series expansion, it is easier to make the series converge to the function // at these two 


Furthermore, we can delete from this set either those with odd n or 
those with even n, depending upon the boundary condition at the right end. 
If the right end is also held fixed, that is, 

u(L) = 

to satisfy this imposed boundary condition, we must choose n as even. 
If the right end is allowed to move (not necessarily free), we must choose n 
as odd, so that the sine series can converge at x = L to a value other than 

Let us suppose that at the right end a concentrated mass M is attached. 
The boundary condition there is 

, du d 2 u 

k— = -M— at x = L (101) 

ox ot A 

in which k = EA. Evidently this condition is not satisfied by 

. YlTTX 

sin — n = 1, 3, 5, • • • 

But this is not important for reasons to be seen later. 
The kinetic energy rand potential energy Kfor this system are given by 

1 [ L (du\ 2 , 1 (du\ 
If we represent u(x, t) by 


-) [.. \ 'At ! ' '") \ 'At I 

x = L 

i r L /du\ 2 

V=- I k\ — \ dx 


w= |^ Wsin (?Lzi^ (1 o3) 

?=i IL 

we can obtain dujdx by termwise differentiation, 

du " (2i-l)7T (2i-l)7TX ritXA ^ 

~-=I qlt) cos —L (104) 

ox 1=1 2L 2L 

which is valid for all < x < L. We can also obtain dujdt by termwise 


du " e (2i- \)ttx 

-= StfXOsin - lr (105) 

ot i=\ 2L 


assuming that the resulting series is uniformly convergent with respect to t. 
By substituting (104) and (105) into (102), we obtain 

1 CO CO 

Z i = 1 3 = 1 


i oo co ' 

Z i = l j = l 


f L , , • (2/ - 1)ttx . (2/ - 1)ttz , . . 

a tj = m(«) sin — sin dx + {-\)'- J M 

Jo 2L 2L 

c „ = JV)^(2/-l)(2y- l)cos ( ?^ 7 l^cos^ z ^-*. 
Now the conservation of energy requires that 

^(r+ k) = o 

CO oo 

2 2 (««& + ctfMt = o 

i = l j=l 

Since g's are supposed to be independent, 


2 arf, + c ijqj = i = 1, 2, 3, • • ■ (107) 


Thus we have an infinite set of linear differential equations of an infinite 
number of unknowns. 

Whether or not such infinite formulation is always mathematically 
legitimate is too involved a question to be discussed here. For a given 
problem this question may be answered when the solution is obtained and 
each step is retraced to examine its legitimacy. Even that task is difficult. 
Fortunately, as subsequently demonstrated, the reasoning used here leads 
in practice to the same result as the Rayleigh-Ritz method, of which the 
mathematical foundation is more secure. 

We note that each of the function <j> t in (99) satisfies the imposed 
boundary conditions, hence can be used as a generating function in the 
Rayleigh-Ritz method. If we terminate (99) at the /2th term, the energy 
expressions (106) will be that of an ^-degree-freedom system. The fre- 
quency equation obtained from (107) with i,j= 1,2,..., //, is exactly that 
for minimizing (96). 




As an example, let us take the simple case in which m and k are con- 
stants in (102). The resulting problem was solved exactly in Art. 4.3(a); 
therefore, we can evaluate the accuracy of the method. Let 

n = 4 




. TTX ZlTX 5tTX IttX 

u(x, t) = ft sin 2l + q2 Sin ~2L +%Sm ~2L + q * Sm ~T 

By putting this into (106), we obtain 

T = \mL[qf + q 2 + q 2 + q 2 + (q x - q 2 + q 3 - q^f] 
= \mL(2q^ + 2q 2 + 2q 2 + 2q 2 - 2q x q 2 + 2q 1 q 3 
- 2q x q± - 2q 2 q 3 + 2q 2 q^ - 2q 3 q 4 ) 

}*E« + 

, f i + 25 ft « + 49? 4 2 ) 


A = 

2 -1 

1 2 

1 -1 

1 1 




c = 











■1.0000 1.0000 -1.0000" 

0.2222 -0.1111 0.1111 

■0.0400 0.0800 -0.0400 

0.0204 -0.0204 0.0408 

4mL 2 

By the matrix iteration method described in Chapter 3, we obtain 



\M6V(klm)L = \M6\/(EI P )L 


r, = 92.8 sin 5.68 sin h 1.99 sin sin 

1 2L 2L 2L 2L 

The exact solution of (24) for jli = \ is 

o h = l.0169V(Elp)L 


r x = Bs'm 1.0769- 

in which B is an arbitrary multiplication constant. The comparison 
between i\ and T\ is shown in Fig. 93. 




Note that the approximate function /\ does not satisfy the boundary 
condition (101). This is reasonable, since that boundary condition 
describes an equilibrium relationship that supplements the differential 
equation of motion. In other words, both the differential equation (5) and 
the dynamical boundary condition (101) originate from Newton's law. 
In the infinite-series formulation they are replaced by (107). An approxi- 
mate solution of (107) is not expected to satisfy (101) any more than it is 
expected to satisfy the differential equation of motion. 23 However, by 

Figure 93 

taking a sufficient number of terms in (103), the two sides of (101) can be 
made arbitrarily close to each other at x = L — e, where e is arbitrarily 

(b) effects of constraints on natural frequencies 

A very important concept in vibration analysis is that the introduction 
of additional constraints into a system will raise all the natural frequencies 
of the system (or at least will not lower them). For instance, in the lateral 
vibration of bars all the natural frequencies will be raised if a free end is 
made into a hinged end, and they will be raised again if the hinged end 
becomes a built-in end. 

To prove this statement in a general way, we utilize the maximum- 
minimum characterization of eigenvalues discussed in Art. 4.6(c). Before 
we proceed, let us point out that an imposed boundary condition, being 
geometrical in nature, constitutes a constraint, whereas a natural boundary 

23 In this particular example the approximate solution does satisfy the differential 
equation of motion. This is an exception rather than a rule. 


condition is not considered as a constraint, since it originates from con- 
sideration of forces. A constraint may also be any interdependence 
relation imposed upon the coordinates of a system; for instance, such 
imposition as 

w(0, t) = 3w(L/2, /) = u(L, t) 
or in (99) 

<7i = % = <?5 = ' ' • = 

Let S be a vibrating system and S be another vibrating system, which 
can be obtained by imposing additional constraints on S. The two 
systems therefore share the same differential operators L and M, and their 
energy expressions derived from an (essentially) admissible function will 
also look the same. Hence Rayleigh's quotients for the two systems are 
computed in the same way. We note also that since any constraint of 
system S is also a constraint of S, but not vice versa, any admissible or 
essentially admissible function of § is also admissible for 5, but not vice 

Let r l9 r 2 , . . . , r fc _i be the first k — 1 eigenfunctions of S. Let C k be the 
class of functions that is admissible to S and orthogonal to t\, r 2 , . . . , r k _ l9 
with respect to M. Let C k be the class of functions that is admissible to S 
and orthogonal to r ls r 2 , . . . , r k _ v It is clear that every function in C k 
must also belong to C k , but not vice versa. According to the minimum 
characterization of eigenvalues, the function in C k having the least 
Rayleigh's quotient is the kth eigenfunction of S, namely r k , and 

fi(W) = <■>* 

Since C k contains all the functions in C k , the minimum value of Rayleigh's 
quotient formed by a function u in C k is no less than Q((r k )). 

lOmirtn > 2((0) = < 

In the meantime, if co k 2 is the kth eigenvalue of S, according to Art. 

*V > [Q((u))] mm 

because C k is formed by removing from functions admissible to Sail those 
which do not satisfy 

r.M(w) da = ] = 1, 2, • • • , (k - 1) 

and, insofar as S is concerned, r l9 r 2 , . . . , r k _ 1 is a set of arbitrary functions. 

c5 7c 2 >fo fc 2 (108) 


The natural frequencies obtained by the Rayleigh-Ritz method can also be 
considered as the true natural frequencies of a hypothetical system, which 
is obtained by imposing additional constraints to the actual system. 
By terminating the infinite series (99) at the nth term, we introduce the 

q n+1 (t) = q n+2 (t) • • • = 

We note also that in proving (108) all we needed was the fact that the class 
of admissible functions of S is a smaller class than that for S. It is therefore 
clear that the n natural frequencies obtained by the Rayleigh-Ritz method, 
interpreted in the light of either Art. 4.7 or Art. 4.8, must be equal to or 
greater than the first n frequencies of the actual system. Hence 

and by retaining more terms in (99) or by adding more functions to the 
generating set for (82) the answers tend to be improved. In other words. 
the Rayleigh-Ritz method approximates the natural frequencies from above 
and can always be used to establish their upper bounds. 


(i) From the simple example worked out in (a) it is seen that the energy 
method can give usable engineering answers for the lowest natural fre- 
quency with a relatively small, well-chosen generating set. The eigen- 
function obtained from the approximate method is, however, not quite so 
reliable, especially at places near a boundary point at which a natural 
boundary condition is not satisfied by the functions in the generating set. 
A look at Fig. 93 will reveal this fact. 

(ii) If n lowest natural frequencies are to be found, a generating set 
larger than n functions is needed. How much larger the generating set has 
to be depends on the nature of the functions chosen. If they have close 
analytical resemblance to the eigenfunctions of the problem, only a few 
more than n functions are needed. The existence of such resemblance can 
be assured if the diagonal elements of both A and C for the approximating 
system are much larger than their off-diagonal elements. Although this 
condition is not mathematically necessary for the generating set to 
generate functions closely approximating eigenfunctions. in practice we 
always aim to achieve this condition. 

(iii) It must be pointed out that in practical problems we are seldom 
interested in knowing more than the first few natural modes of the system. 
Furthermore, although the energy method combined with a digital 
computer can determine accurately the higher eigenvalues and their 
associated eigenfunctions, they may not describe the actual modes of 


vibration. This is because the mathematical formulation often loses its 
physical validity for these modes. For instance, the first few modes of the 
lateral vibration of a round bar having a length-to-diameter ratio of 10 can 
be described accurately by the equations or energy expressions obtained 
from the elementary theory of beams. But these same equations and 
energy expressions are grossly in error when applied to, say, the tenth or 
higher modes. 

(iv) The labor required as well as the accuracy obtainable in solving 
problems approximately by the energy method depends upon the choice of 
generating functions <f> x , </> 2 , . . . , <j> n . A desirable choice is one which 
facilitates the evaluation of the integral in (83) and yields a pair of matrices 
A and C, whose diagonal terms are much larger than the off-diagonal 
terms. In other words, we aim to make the </>'s and the true eigenfunctions 
as nearly alike as possible, without excessive computational labor. 

There are no hard and fast rules to be followed in achieving these 
results. The following observations, however, may serve as a rough guide. 

(i) Polynomials are often conveniently chosen as generating functions 
because they can be easily differentiated, integrated, and made to satisfy 
the boundary conditions. For instance, in analyzing a nonuniform beam 
with a built-in end at x = and a free end at x = L the generating set can 
be simply 

/yiLt /yd /y**± ■ • • /y»'t 

which satisfy the imposed boundary condition at x = but not the natural 
boundary condition at x = L. Or we may use as the generating set 

fa = x 4 - 4Lx s + 6L 2 x 2 
fa = x 5 - \0L 2 x* + 20L 3 * 2 
fa = x 6 - 20L 3 x* + 45ZAr 2 
fa = x i - 35L 4 x 3 + 84LV 

which satisfy all the boundary conditions. 

Tt is to be expected that for a given accuracy we need fewer functions 
of the second set than will be needed if the first set is used. The matrix 
problem resulting from the second generating set will be smaller but it will 
take longer to compute a u and c u defined by (83). 

(ii) It is always desirable to take advantage of any known orthogonal 
properties relative to the differential operators M and L. For instance, if 
M is merely a simple multiplication constant, then the generating set 
can be simple sine and cosine functions satisfying the imposed boundary 


conditions. In that way, a {j = 0, when i^=j; or, if M = x, we may 
choose Bessel functions of the appropriate order satisfying the imposed 
boundary conditions. 

(iii) If by changing the natural boundary conditions of a problem the 
eigenfunctions are known, these eigenfunctions can be advantageously 
chosen to be the generating functions of the original problem. For example, 
the eigenfunctions (36) of the cantilever beam shown in Fig. 88 are good 
generating functions for the system in Fig. 92, especially when the right- 
end spring supports are soft. Similarly, if by a slight change of the 
differential equation the eigenfunctions of the problem are known, they 
can be used as the generating functions of the original problem. 

4.9 Forced Vibration of Elastic Bodies 

(a) concentrated force applied at a fixed point 

In Chapter 3 we discussed a method of solving forced-vibration problems 
by a transformation of coordinates that results in a set of differential 
equations, each of which contains only a single unknown, namely, one of 
the principal coordinates. The same approach has proved to be a con- 
venient one for the problem at hand. 

The essence of the discussions in Arts. 4.7 and 4.8 is that the distribution 
of elastic force and that of the inertia force in a vibrating elastic body can 
be characterized either by two differential operators L and M. together 
with a set of boundary conditions, or by two energy expressions Fand T. 
Furthermore, if u is the actual displacement function during vibration, 
the energy expressions and the differential operators are related by 

= *J> 

M(w) da 


V = \ I ul(u) da 

J 2 

where u stands for dujdt. 

In free vibrations the principle of conservation of energy 

l(T+ V) = 

leads to the equations of motion. Similarly, we may derive the equations 
of motion for forced vibrations from 

— (T + V) = rate of doins work by external forces 


For a given elastic system the energy expressions V and T, as well as the 
operators L and M, are the same regardless of whether the vibration is 
forced or free. 

Let u be the displacement function describing the forced vibration of an 
elastic body. Let the eigenfunctions for the free vibration of this elastic 
body be r lt r 2 , . . . , r,-, ... . The function u can be expanded into an 
infinite series of r's. 

<x, t) = fp#Mz) (110) 

i = \ 

We assume here that there is only one space variable, x. However, the 
generalization to cases with more than one space variable is immediate. 
As the eigenfunctions in free vibration, the r's have the property 

f r,M(r 3 ) dx = 

i^j (HI) 



L(r,) = a>/M(r,) (112) 


Now let us call for simplicity 

rW da = m u (113) 


By substituting (110) into (109) and utilizing (111), (112), and (113), we 

oo oo 


d °° 

- (T + V) = 2 m^ + w,.^ 

a / { = i 

Let a force/(?) in the direction of u be applied at a fixed point in the body 
at x = £. The rate of doing work by this force is 


Therefore, the energy equation leads to 

i [m«(A + o>i 2 Pi) - foymipi = o (H4) 




Since p/s are not dynamically related to p t or p i9 being dependent upon 
initial conditions which are quite arbitrary, the vanishing of (1 13) demands 

m u (pi + cofc) =f(t)r i (i) i = 1 , 2, 3, • • • (115) 

There is an infinite number of equations in (115), but each contains only 
one unknown, which can be found separately by the methods discussed in 
Chapter 1 . The initial conditions for p i are derived from those for u. 


i = i 

\Ot/t = t i=l 

After the/?'s are obtained, they are put into (1 10) to get the solution for u. 

Figure 94 

The solution is in the form of an infinite series. This series must be 
convergent and termwise differentiable in order that the operations leading 
to (114) legitimate. Such is usually the case. 

To illustrate the procedure, let us take the case of a simply supported 
beam, having uniform cross sections and subjected to a concentrated force 
f(t) at x = £, as shown in Fig. 94. According to the results on p. 257 
the eigenfunctions for this case are the sine functions 


r,- = sin 

and the eigenvalues are given by 




t _EW_EI/*i\ 
1 " P A " pA\l) 

« = f " 

2 f7TX A L 

sin' 2 — — ax — — 
2 2 

for all i 


By substituting these into (115), we have 

EI /tt\ 4 a 2 . fai ., 

P A 


The results given in (a) can now be generalized in the following three 

(i) If the external force is distributed over the elastic body with the 
distribution described by 


(115) is modified into 

mJPi + cofc) = f /({, t)rm dS (116) 


by the principle of superposition. 



Figure 95 

(ii) If a concentrated external force changes its point of application, 
then £ = £(/), and (115) is changed into 

m u (p t + o>* Pi ) =f(t)rlHt)] 

This is true because there is nothing in the derivation of (115) that implies 
I is fixed. This generalization is of practical interest in studying vibrations 
of a structure under a moving load. For example, in Fig. 95 a weight W 
moves across the simply supported horizontal beam with a uniform 
velocity v, and the force exerted on the beam is of magnitude 

-W--a u ,=f(t) (117) 


in which a w is the vertical acceleration of W. If a w <^:g, (115) is modified 

mdpi + w«Vi) = - Wrgvt + f )] 


in which £ is the location of W at t = t . Otherwise 

(d 2 u „ d 2 u av\ 


i = i 

By substituting this expression for a w in (117) and (115) and replacing £ 
by (*;/ -f | ), we obtain a set of linear differential equations for /?'s in 
which the coefficients vary with time. Furthermore, all the unknowns 
appear in all the equations. Such equations are not readily solvable. They 
are mentioned here merely to illustrate the fact than when additional 
inertia force due to Sexists the original set of eigenfunctions is no longer 
that of the new system, and since this additional inertia force changes its 
location the system is no longer time-invariant. 

(iii) Generalized forces. In the preceding examples we assume that 
work is done by external forces applied in the direction of the displace- 
ment w, and for a single concentrated force at x = £ the rate of doing work 

d ,„ . _ „_Jdu\ 

A (r + K)=/(0L 

But there are other ways in which external forces may do work. Taking 
the lateral vibrations of bars as an example, we know that work can be 
performed by a couple or moment in the plane of bending of the bar. In 
that case 

d ._ __ _..Jl d 2 u 

(r+K)-«(0(^) -S^Or/WO 

Thus (115) is modified into 

mdpi + co 2 Pi ) = M(t)r^) 

Work can also be performed by a pair of equal and opposite axial 
forces acting on two ends of the bar, as shown in Fig. 96. Let the force be 
P(0; then 

1{T+ V) = P(t)X(t) 

where X(t) is the displacement relative between the two ends. To compute 
X{t), we observe that two points on the neutral axis a distance dx apart 




are separated in the axial direction during vibration by a distance dx cos f), 
where Q is the slope of the deflected bar. 24 Hence 

d C L d C L 6 2 

X(i) = - dx (1 - cos 0) = — dx — 


Jo 2\dxJ 



dX = dx(l -cos 0) 

U-dx J 

Figure 96 

By utilizing (1 10), we have 

*c) = ii piPi f^'wc*) * 

i = 1 i = 1 Jo 

This modifies (1 15) to read 

™u(pi + <o*Pi) = P(t) 2 Pi r/(xy/(x) dx 

i=i Jo 

The introduction of axial forces is similar to a change in the elastic 
property of the system. As a result, the r's are no longer the eigenfunctions 
and the/?'s no longer appear separately in the equations. In the particular 
case of a uniform bar with two hinged ends 

. 17TX 

r£x) = stn — 

, in 17TX 

r„ (x) = — cos — . 
1 W L L 

it so happens that 


r/(xy/(x)dx = i^j 


o 2L 

24 The effects of the direct compressive stress produced by P(t) are neglected. 


Here again we have a set of equations with variables separated: 

*2 2 


m u pi + 1%^ - ^ ?(*)) Pi = 
Since for this particular case 

we have 

H = M sin 2 l ^dx = ?-— 
Jo L 2 

/O^L i 2 7T 2 (i 2 7T 2 EI 

If P is a constant, the foregoing equation can be considered as the standard 
form for systems in free vibration. 

Pi + &*Pi = ° 


i 2 7T 2 i 2 7T 2 EI \ 

P AL 2 \ L 2 ) 

ojs = 

In other words the existence of an axial force merely modifies the elastic 
force of the system; hence it changes the natural frequencies. It is inter- 
esting to note that if P is equal to Euler's critical load for column buckling 

7T 2 EI 

P = -[? 

then e^ = 0, and the system is in a neutral equilibrium. If P is less than 
Euler's critical load, all the natural frequencies will be real and positive, 
and the system is stable. If P is greater than Euler's critical load, one or 
more of the natural frequencies will be imaginary, and the system is 
capable of a spontaneous motion in which the displacements will grow 
without limit. The system is therefore unstable. The fact that a column 
subjected to axial forces greater than Euler's buckling load is an unstable 
column is thus explained by theory of vibration, whereas it cannot be 
explained by elementary column theory. We can therefore state that the 
criterion for the stability of an elastic system is the absence of negative 
eigenvalues or imaginary frequencies in the free vibration problem of the 



The indicial response of an elastic system can be defined as the vibration 
caused by a concentrated unit step force applied at a fixed point in the 
system and at a time when the system is at rest. 

Let this indicial response be denoted by the function £/(£, x, t) for a 
one-dimensional elastic system. If its eigenfunction-series expansion is 

U&x,t) = fplt)rlx) 

i = l 

then p i is the solution to 

m ii {p i + M ?p t ) = S(t)r^) i = 1, 2, 3, • • • 
under the initial conditions that 

P M = o /),(0) = o 

According to (75) of Chapter 1, 


PAD = -^- 2 (i - cos «v) 

m u cof 

i=i rn ti (o t 2 

The impulse response, which describes the free vibration caused by an 
impact, is obtained by differentiating U with respect to t. 

Thus, if the system is initially at rest and is subsequently set into motion by 
a force /(0 applied at x = £, the solution is 

00 r (E\r (x\ C f 
u(x, 0=1 - — /(t) sin co t {t - r) dr (118) 

i = i m ii p i Jo 

And, if we are dealing with distributed force of distribution g(£, t), the 
solution is 

u(x, t) 

Z sin co t (t — r) dr d$ (119) 

i-i Jo Jo rnuoii 

This is the general solution of forced vibration of a one-dimensional 
elastic body which is initially at rest. 


4.10 Vibration of an Infinite or Semi-Infinite Elastic Body — 
Wave Phenomenon 

(a) continuous frequency spectrum 

The systems we have dealt with so far are finite in extent. A system of 
this kind was found to possess a set of distinct natural modes, and its 
natural frequencies form a discrete set of values. As a result, any free 
vibration it may have is representable by an infinite series 


m = 2Q cos (cos — <x l )r i (120) 

The total energy possessed by the system in a given vibratory motion is 
T+ V= U [uM(u) + ul(u)]da 

J 2 

= i!m ii C i W (121) 

i = l 

where m u is defined in (113). The total energy expression (121) indicates 
that a unique and definite amount of energy is associated with each natural 
frequency or that between different natural modes there is no energy 
transfer. Because the natural frequencies are discrete the free vibrations of 
a finite body are said to have discrete frequency spectra. 

To examine the changes when an elastic body becomes infinite or semi- 
infinite in extent, let us reflect that for a finite body the eigenvalues or the 
natural frequencies are determined by the conditions at the boundary. 
When a system becomes infinite or semi-infinite, all or part of its boundary 
is located at infinity, where the prevailing conditions are hardly expected 
to have an effect on the motion of the interior mass elements of the 
system. Take, for example, the longitudinal vibration of a uniform bar 
with a fixed end at x = 0. The eigenfunction is 

r(x) = sin fix 

The allowable values for (3, or the eigenvalues, are determined by a linear 
homogeneous boundary condition at x = L. 

ar + br' = a sin /3L + bfi cos pL = (122) 

This becomes the characteristic equation of the problem. Now. as L 
becomes infinitely large, this equation can be satisfied by any arbitrary 
real value 25 of fi. Looking at the problem in another way, we see that the 

25 To state it correctly, we should say that given an arbitrarily small e > 0. with a 
sufficiently large L we can find for every real positive fi a ft such that /3 satisfies (122) 
and \p - j5| < e. 


characteristic equation for a one-dimensional continuum is satisfied by 
PL = (one of a set of dimensionless numbers). The difference between two 
consecutive numbers is always finite. 

Pi+iL ~ PiL < N < oo for all i 

lim (/?, +1 - ft) = for all i 

L— > oo 

All this means that when an elastic body is infinite in extent its frequency 
spectrum becomes continuous. 

(b) a phenomenon of wave propagation 

The concept of the natural mode of vibration loses its physical signi- 
ficance in dealing with an infinite or semi-infinite body, not only because 
the natural frequencies are no longer distinct, but also because the vibra- 
tion represented by an eigenfunction cannot be excited without an infinite 
amount of energy input. For instance, a longitudinal vibration represented 

u(x, t) = C sin fix cos (cot — a) 

p = coVe/p c>° 

is mathematically possible for a uniform bar of infinite length. But such 
vibration is of little physical interest, since its energy content is infinite as 

1 f °° 
(J + V) = - co 2 C 2 sin 2 Bx dx = oo 


A more realistic physical problem follows. An infinite elastic medium is 
initially at quiescent state. A finite disturbance is introduced at certain 
parts of the medium. What is the resulting motion of the medium? 

Consider again the longitudinal vibration of a uniform bar, which is 
governed by the one-dimensional wave equation : 

d 2 u 3 2 u 

8?= C *W (123) 


c = + Ve[p 
The general solution to this second-order, partial differential equation is 
u = (f)(x - ct) + ip(x + ct) (124) 


in which <j> and tp are two arbitrary, twice-differentiable functions. The 
way x and t enter into the arguments of <f> and ip indicates that a propaga- 
tion phenomenon is involved. Obviously, 

c/)(x 1 — cti) = <f>(x 2 - ct^) 


x 2 - x 1 = c(t 2 - tj) 

Stated in words, these equations say that whatever exists at x Y will exist at 
x 2 at an instant (x 2 — x^/c later. The phenomenon represented by <f> 
therefore propagates to the right. Similarly, 

ytei + ct Y ) = yj(x 2 + ct 2 ) 

x 2 -x 1 = -c(t 2 - tj) 

which says that whatever exists at x 1 existed at x 2 at an instant (x 2 — x^)jc 
earlier. The phenomenon represented by ip therefore propagates to the 
left. In either case the speed of propagation is the constant c. 

Now let us suppose we have a very long bar extending from x = a < 
to x = b > 0. A small external disturbance is introduced into the bar 
near x = at t = 0. Let this disturbance be in the form of an initial 
displacement, which is restricted to a small interval containing the point 
x = 0. The initial conditions are thus 

u (x, 0) = U(x) 



As shown in Fig. 97, the initial displacement function L\x) is zero, 
outside of a small interval. It can be readily verified that both the differ- 
ential equation (123) and the initial conditions (125) will be satisfied if we 

in (124), or 

u(x, t) = ±U(x- ct) + \U(x -f ct) (126) 

Hence the disturbance introduces splits into two equal waves and pro- 
pagates in two opposite directions with a constant speed c. If the bar 
extends to infinity at both ends, then (126) is valid for all r's. If, on the 
other hand, the bar terminates at a finite point or points, the solution 
(126) is valid for a time interval < t < t l9 in which r x is time required for 
the disturbance to reach one or both ends of the bar. This is so, regardless 




Free end 






l^2e-^ U — 2e — J 

Fixed end 


Figure 97 

Figure 98 


of the end conditions, because, according to (126), until the disturbance 
reaches an end the mass element at that end has zero displacement and 
zero velocity, a combination that satisfies any homogeneous boundary 

If the bar has an end, the disturbance wave will be reflected as it reaches 
that end. In a certain subsequent time period the reflected wave and the 
incident wave coexist in the bar. Afterward, only the reflected wave 
remains. The manner in which the reflection takes place depends on the 
end condition. At a fixed end 

"incident ' "reflected " 

and at a free end 

incident \ OX] reflected 

These conditions are graphically illustrated in Fig. 98. Because neither 
a fixed end nor a free end can absorb energy, the reflected wave has the 
same shape and energy as the incident wave. 


Consider now a bar having fixed ends at x = —L/2 and x = L/2. 
An initial disturbance is introduced into the bar. This disturbance splits 
and propagates toward the ends with speed c. It is evident that each of 
the two parts will return to its original location, shape, and direction of 
propagation after being reflected once at each end. After an elapsed time 
of 2Ljc, the two parts reconstitute the original disturbance. This elapsed 
time thus becomes the fundamental period of the natural vibration 

t = — = ilV^Je 


If the bar has a fixed end and a free end, each of the split disturbances will 
have to be reflected twice by each end to return to its original condition. 
The period is then 

T = — = ALVpjE 

Take a special case in which the initial disturbance for a bar having two 
fixed ends is given by 


U{x) = C sin — 




Referring to Fig. 99, we see after the wave splits that the reflection from 
one part complements the incident wave of the other part to constitute a 
shifted half sine wave. Hence for this special case (126) is always valid. 

7T 77" 

u(x, t) = |C[sin - (x — ct) + sin - (x + ct)] 

J-/ Li 


= C sin — cos — 

Li L-d 

t = 

t = t 


Figure 99 

Since the shape of the displacement curve is now always a half sine wave, 
we call it a standing wave, and it represents the first natural mode of 
vibration. A standing wave can also be formed in this case by 

U(x) = C sin 

in which n is any integer. 


We now illustrate another approach to the analysis of vibrations of an 
infinite body. A mathematically rigorous development of the subject to be 
discussed must utilize the theory of Fourier transformation. Here we have 


to be satisfied with a version that is mathematically less rigorous but 
physically more informative. 

Let us begin with a finite bar extending from x = — L/2 to x = 12. 
According to (37), the free vibration of this bar can be described by 


v(x, t) = 2 C, cos (a),/ - a^rjix) (37) 

Since for each i there is a ft, we may consider C, co, a, and r as functions 
of a variable ft Let us rewrite (37) as 

v(x, t) = 2 -rf cos (co,/ - a,K(*) A/?, (127) 

*-i Aft 

where Aft = ft — ft_ r We noted in (a) that as L -+ co, ft -+ 0, Aft -> 0. 
and Q-+0. If the limit 


lim —4- = lim £, < co 


exists, the right-hand side of (127) becomes an improper integral as L -> x 
or Aft-^0. 

*;(*, 0=| £(/?) cos {cot - a>(ft a;) dfi (128) 



co = pWEJjpA (129) 

This solution is arrived at strictly in a formal way. We have no knowl- 
edge of the appearance of E(ft\ oc(ft, and r(ft x), nor do we know whether 
or not the improper integral converges. 

To find out how (128) can be made to fit the problem, we observe first 
that if the partial differential equation (28) is to be satisfied by (128) 
r(ft x) must satisfy (30). Hence from (32) 

r(ft x) = A sin fix + B cos fix + C sinh ftc + Z) cosh ftr 

in which A, B, C, and D may be functions of ft but not of .r. Since (128) 
must converge for all x and since hyperbolic functions grow without limit, 
the coefficients C and D must vanish. Thus 

r(ft x) = A sin fix + B cos ftz 

= F(ft cos (/5k - 6) 

By substituting this into (128) and combining E and .Finto a new function 
G, we have 

v (x, 0=| G(/?) cos (ft)/ - a) cos (ftr - d) dfi (130) 


Now let us take a simple initial condition 

v (x, 0) = V(x) 

Furthermore, let us assume that Vis an even function, V(x) = V(—x). It 
is not difficult to see that these conditions imply 

<x(/?) = and d(p) = 

K(s) = I °°G(/5) cos fix dp (131) 


This is the formula for a Fourier cosine transform, 26 and the inverse is 

given by 

2f co 
G(P) = - V(x) cos (fix) dx 

The solution is then 


x,t)= G(P) cos cot cos fix dp (132) 


where co is given by (129). By an identical procedure, we can show that if 
the initial displacement is an odd function of x 

v (x,0) = V(x) = -V(-x) 

d = -tt/2 


v(x, t) = G(P) cos cot sin fix d/1 (133) 



G(f$) = - I V(x) sin ^x dx 

77 Jo 

Since any arbitrary initial displacement function can be decomposed into 
an even function and an odd function by 

V(x) = i[V(x) + V(-x)] + \[V(x) - V(-x)] 

by the principle of superposition, we can obtain the solution for any 
initial displacement V(x). The result is 

77 Jo J- 

v(x, t) = - V(i) cos 6(£ - x) cos cot d£ dp (134) 

26 See Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951, p. 1! 


4.11 Method of Finite Difference 

A very straightforward way of solving an eigenvalue problem numerically 
is by the method of finite difference. The principle is simply that differen- 
tial equations and integral equations can be approximated by algebraic 
equations. The technique, however, has many refinements and ramifi- 
cations. In this article we shall illustrate the basic principle by simple 
examples. 27 

(a) finite difference approximation for a differential equation 

Consider the eigenvalue problem in the lateral vibration of uniform 
bars. Formulation by differential equation leads to 

S-^-0 (135) 

Let (n + 1) equally spaced points be chosen along the bar. These are 
called the pivot points: x = 0, x x = h, x 2 = 2/z, . . . , x n = nh = L. Let 
the corresponding values and the derivatives for r be designated by 

r = r(0), r x = r(h), r 2 = r(2h), ■•-,/•„ = r(L) 

r ' = r'(0), r{ = r\h\ • • • , r n ' = r\L) 

For the purpose of deriving formulas we must also introduce the 
following intermediate pivot points: 

_h _3/z 

Xl A ~ 2 Xiy * ~ ~2 etC ' 

The derivatives of r of various orders can then be approximated by the 
following difference formulas 

hh" = h(r' i+1A - r\_ v ) = {r i+1 - r t - r< + r^J 
= r i+1 - 2r L + r t ._! 

= r i+H - 2r i+H + r^ H - (r i+1A - 2r t _^ + r t _ SA ) 

= r i+H - 3r ( +y 2 + 3r { _ 1A - r i _ S/i (136) 

It is not difficult to show that the formulas resemble that of binomial 
expansion for 

a%a A - a~ 1A ) j 

27 See also the example at the end of Art. 4.2(a). 


in which the exponents of a become the subscripts for r and the coeffi- 
cients in the expansions are the same. At any rate, the approximation for 
the fourth-order derivative needed for the present problem is 

d*r „„ 1 

^a = r i = ^i fo+2 ~ 4r i+i + 6r i ~ 4r i-i + r i-2) 

/= 1,2, ••-,(/!- 1) 

We can thus approximate (135) by 

r i+2 ~ 4^+i + 6r t - 4r,_ ± + r,_ 2 = (hpfr, / = 1, 2, •••,(«- 1) 

There is one such equation for each interior pivot point. 

Besides the interior pivot points, there are two boundary points. 
Moreover, to write the difference equation (137) for an interior pivot 
point near the boundary, we shall also need two imaginary pivot points 
lying outside the interval to L. The values of r at these four points must 
either be known or be expressed in terms of those for the interior points 
by the use of the boundary conditions. 

At a hinged end, 

or r , = 



thus r = 

r"(0) = thus 


- 2r + r x = 

At a built-in end 



thus r = 



thus r x — r_ 

At a free end the situation is slightly complicated. There we try to 
write (137) differently for the nearby interior pivot points so that they 
will not include r_ x and/or r . This is done by the following scheme. 

The fourth derivative of the first pivot point may be written 

Since r 

— (r 3 - 2r 2 + r x - 2r 2 + Ar x - 2r ) 

j£ (r s - 4r 2 + 5/-J - 2r ) 


In the meantime, the same derivative can also be written 

* / in Wx *■ 

Th (2 ~ t{i) ~ih 

/ /// lll\ y III III r\ 

2 - r o ) = —,. r 2 smce r = 

where we may approximate r% by 

r" — Ur'" 4- r'" \ 
r 2 — 2V2.b + r l.5>> 

Utilizing (136), we obtain 

(r 4 - 2r 3 + 2r x - r ) (139) 

4/z 4 
Elimination of r between (138) and (139) yields 

r'i = ^ (2r, - 5r 3 + 4r 2 - rj = /?% (140) 

This equation now replaces the first equation in (137). Also by equating 
(138) and (139), we can express r in terms of r ls r 2 , r 3 , and r 4 . The result- 
ing expression can be used to eliminate r in the second equation of (137). 

Example: To find the natural frequency for the first mode of lateral 
vibration of a uniform bar with hinged ends. 

Let the spacing of pivot points be 


Because of the symmetry of the motion to be investigated and of the 
end conditions it is known that 

r = r 6 = r_j_ = r t = r 5 = r 1 and r 2 = / - 4 (141) 

By utilizing these conditions, we obtain the first three equations for 

5/i - 4r 2 + r 3 = Ar x 

-4f! + 7r 2 - 4r 3 = Ar 2 (142) 

2r 2 - 8r 2 + 6r 3 = Ar 3 

\ 4 


The remaining two equations of (137) are duplicates of the first two. The 
lowest eigenvalue of (142) is found to be 

I = 0.0718 

and the corresponding eigenvector is (1, 1.732, 2). Hence 

0L = 6 x 0.0718^ = 3.108 


The true answers are known to be 

fiL = 77 = 3.1416 


r x : r 2 : r 3 = sin — : sin - : sin — = 1:1 .732 : 2 
6 3 2 

This represents an error of about 1 per cent in {$ and 2 per cent in oj, since 
oo is proportional to /? 2 . The values for r at the pivot points happen to be 

It is interesting to note that this method of approximation results in a 
frequency value lower than the true value, whereas most other approxi- 
mating methods give a value higher than the true value. This is because 
the two sides are not equal in the approximation (136), and 

/?%" = (r i+2 - 4r, +1 - 6r, - 4/V-i + r t _ 2 ) + e 

The error e is approximately 28 

In a problem in which r is of the opposite sign to r" the error term makes 
the finite difference approximation to r"" too small. Physically, it means 
that the restoring force based on the approximation is too small or the 
system is too soft. The result is a frequency lower than the actual. 

(b) finite difference approximation for integral equation 

In Art. 4.5 it was shown that the eigenfunction r(x) satisfies the integral 

— 2 r(x) = I G(x, !)m(!)r(£) d£ 

00* Jo 

If we choose a set of equally spaced pivot points and integrate the right- 
hand side by either the trapezoid rule or Simpson's rule, we obtain a set 
of algebraic equations in r-values at the pivot points. Let us take again 
the vibration of a uniform bar with hinged ends and divide the interval 
into six segments. For a generic pivot point i the trapezoid rule yields 

~AoJ 2 = h \2 Gi ° r ° + GilKl + ° i2r2 + Gl * rz + G * 4 '' 4+ Gi * r * + 2 Gl * r *) 
For the first mode we can utilize (141) to obtain 

2(G n + G i5 ) ri + 2(G, 2 + G i4 )r 2 + G t3 r 3 (143) 


28 Salvadori and Baron, Numerical Methods in Engineering, Prentice-Hall, New York, 
1952, p. 67. 




It can be shown by theorems in strength of materials that for a simply 
supported beam divided into sixths the influence coefficients are given by 

G„ = 

/z 3 


(6 - i)j(\2i - i 2 - f) if i>j 

Since G, 

G H , this formula is sufficient for computing all G's. Let 
/z 3 



G 12 = G { 

G 22 = 128/7 





^13 = ^31 = ^35 = ^53 = 78 £ 
^23 = ^32 = ^34 = ^43 = 1386 

C7 14 = G 41 = G 25 = G 52 = 626 







Substitute the values for the influence coefficient into (143) and simplify 

42r ± + 69r 2 + 39r 3 = /^ 
69r x + 124r 2 + 69r 3 = Xr 2 
78^ + 138r 2 + 81r 3 = /r 3 


A = 


h* P Aco 2 (f}hf 

This time, since we are looking for the highest / so we can use the matrix 
iteration method, we find 


69 39" 

r l.ooi 



124 69 





138 81 



= 241.7 x 






/ 18 V A 

^ =6 (wj) = 3 - 135 

The error in the natural frequency is thus very small. 



Although the examples we have chosen do not include many complexities 
that appear in practical problems solved by numerical methods, they do 
give an idea of the algebraic and arithmetic processes required. When 
handled properly, all numerical methods presented are capable of giving 
good answers. They differ, however, in convenience for different prob- 
lems. The following general observations can be made regarding the 
relative merits of the two methods presented. 

(i) Difference equations derived from the differential equation formu- 
lation of an eigenvalue problem have the advantage of being easy to set 
up. This is especially true when the boundary conditions are simple and 
when there are no abrupt changes in mass distribution and elastic properties 
in the system. When the number of pivot points is increased for accuracy, 
the number of unknowns in each equation does not increase, although there 
will be more unknowns and more equations to solve. The use of non- 
uniformly spaced pivot points is possible, but it is usually not convenient. 
The smallest eigenvalue of the set of difference equations derived from the 
differential equation corresponds to the lowest natural frequency of the 
system. This is an undesirable feature because it is necessary to compute 
the inverse matrix in order to use the matrix iteration method described in 
Chapter 3. 

(ii) The setting up of difference equations approximating the integral 
equation requires extensive preliminary computation of the influence 
coefficients or the values of Green's function. On the other hand, this 
could be an advantage: (1) when a model of the system is available the 
coefficients can be determined experimentally; (2) the treatment of 
boundary conditions causes few special problems. The integration opera- 
tion is a "smoothing" operation. It is particularly suited for taking care 
of abrupt changes in the interior of the system. For instance, if there is a 
concentrated mass M attached to the midpoint of the bar, its existence can 
be taken care of quickly by adding the term 


to the right-hand side of (143). The difference equation (137) cannot be 
modified so easily. The use of pivot points not evenly spaced requires very 
little modification of the trapezoid rule. When there are abrupt changes 
in cross section, the convenience with which the integral equation can be 
approximated with nonuniformly spaced pivot points becomes a distinct 
advantage. The matrix equation representing the integral equation is 


already in the inverse form. That is to say, the largest eigenvalue corre- 
sponds to the lowest natural frequency so that the matrix iteration method 
previously discussed can be immediately applied to determine the lower 
modes of vibration. 

(iii) In comparing these two methods with the energy method described 
in preceding articles, we can say that in general the energy method is 
capable of much more accurate results if we choose as many generating 
functions as there are pivot points. In the study of two-dimensional or 
three-dimensional systems the number of pivot points required to set up 
difference equations is often too large even for machine computation. 
In that case, the energy method becomes the only feasible one. 


4.1. Set up the differential equations of motion and the boundary conditions 
for the following vibratory systems: 

(a) The longitudinal vibration of a uniform bar, whose two ends are connected 
to a stationary frame by two identical springs which exert axial forces on the 
bar proportional to the end displacements. 

(b) The lateral vibration of a cantilever beam having uniform cross sections 
and carrying a concentrated mass at its unsupported end. 

(c) The swinging of a heavy flexible chain hung by one end from the ceiling. 
The other end is free. 

(d) The longitudinal vibration of a tapered round bar, whose end diameters 
are d and d L . The end with the larger diameter d is built into a rigid wall. 

4.2. The formulation by differential equation of each of the following vibration 
problems consists of two boundary value problems with interrelated boundary 
conditions. Write the differential equations and specify the boundary conditions. 

(a) The lateral vibration of a flexible elastic string initially stretched to a 
tension S. A coil spring of constant K = S/3L connects the point x = L/3 
to a stationary point to one side of the string, L being the length of the 

(b) The lateral vibration of a uniform beam of length L simply supported at 
x = and x = L/3. 

(c) The lateral vibration of a round bar with a built-in end and a free end. 
The bar has a hole which extends from the free end to a depth of L/2 and 
has a diameter of 3J/4, where L and d are the length and the outside 
diameter of the bar, respectively. 

4.3. Obtain by separation of variables the differential equations and the 
boundary conditions governing the eigenfunctions belonging to the systems 
described in Exercise 4. 1 . 

4.4. A uniform bar with a fixed end and a free end is set into longitudinal 
vibration in the following manner. A constant axial force P initially exists at 
the free end of the bar, which is stationary. At / = this force is suddenly 
removed. Determine the coefficient Q in (25). (Note that a, = 0.) 


4.5. Verify the characteristic equations and the eigenfunctions, given in 
Art. 4.3(b) for lateral vibrations of uniform bars having various end conditions. 

4.6. The vibration of each of the prongs of a tuning fork is approximately 
the same as that of a cantilever beam. Design a tuning fork for the note C 
(512 cps), which is to be made of steel weighing 0.286 lb per cu in. and having 
Young's modulus of 29.5 x 10 6 lb per sq in. The effective prong length is 6 in., 
and the prongs are to have a rectangular cross section. What is the frequency 
of the next highest note the fork may put out? 

4.7. A vibrating steel reed having a natural frequency of 60 cps is commonly 
used for the calibration of a stroboscope. If the reed is 2 in. long, what is its 
thickness? Assume that the steel used has the properties given in Exercise 4.6. 

4.8. A simply supported uniform beam of length L carries a concentrated 
mass at its mid-point. 

(a) Show that the characteristic equation for symmetrical modes of vibration 

^0(tan 6 - tanh 6) = 2 

where fi = ratio of the concentrated mass to the mass of the beam 


(b) Find the lowest eigenvalue for // = 2. 

(c) A given function f(x) is to be expanded into an infinite series made up 
of the eigenfunctions of the system. Show how the coefficients of this 
series are determined. 


(a) Show that Green's function for a stretched string with x(0) = and 
x(L) = is 

G(x, £) = (L ~JK for < x < £ 

G(x, £) = ~{L - x) for £ < x < L 

(b) Formulate by integral equation the vibration of a string with a mass 
distribution given by 

/ 7TX\ 

P A(x) = w l 1 - sin — I 

m(x) = 


(a) Obtain Green's function for a uniform bar of length L, which has a built-in 
end at x = and a free end at x = L. 

(b) If the right end of the bar is connected to a stationary frame by a linear 
spring* so that E[ r(L) _ Kr(L) = 

Green's function, different from that in (a), must be used in (68). However, 
the same Green's function can still be used if we modify (68) to read 

1 [ L 

— r{x) = G(x, £)r(|) d£ + f{x) 
<o* Jo 

What is the function fix)'] 


4.11. The static deflection curve of a cantilever beam under a uniformly 
distributed load is 

u (x) = O 2 + 6L 2 - ALx) 

in which C is a constant. The function u(x), which is an admissible function, 
can be used to approximate the first mode of lateral vibration of the beam. 
Compute Q((u)) according to (72) and compare it with a^ 2 determined by (35). 

4.12. The frequency of the first mode of vibration of the beam in Exercise 4.8 
can also be approximated by Rayleigh's quotient for the static deflection curve 
due to a concentrated force at the middle. Show that by the use of (96) and this 
deflection curve we can obtain 

«<*»*-• 48£/ ' 

P ALU/i + 17/35) 

Compare this approximation with the answer to Exercise 4.8(6) for u = 2. 

4.13. Use a three-term Fourier sine series approximation to find the approxi- 
mate answer to Exercise 4.8(6). 

4.14. It has been pointed out that the eigenfunctions for a uniform cantilever 
beam, as given by (36), can be used as the generating functions in the Ravleigh- 
Ritz method for the system shown in Fig. 92. In this procedure it is necessary 
to evaluate 


[r^x)] 2 dx 


(a) Evaluate the foregoing integral for / = 1, 2, and 3, by the numerical 
method. Note that although these integrals can be evaluated analytically, 
a numerical method appears to be less time-consuming, especially since 
tables have been prepared for values of r { {x) by Dana Young and R. P. 
Edgar, University of Texas Publications No. 4913, 1949. 

(b) Use r l9 r 2 , and r 3 of (36) as the generating set to find the approximate 
lowest natural frequency of the system in Fig. 92, assuming that 

K Y = and K 2 = 


4.15. Utilize the results from (a) to find the approximate amplitude of steady- 
state vibration of a uniform cantilever beam at x = L, when a sinusoidal concen- 
trated force 

fit) = pALg cos co f t 

is acting on the beam at x = L. Assume that 2co f = coj = the lowest natural 

4.16. Given a differential equation, 

L(r) = Ir 

and a set of boundary conditions. Green's function associated with this 
boundary-value problem G(x, |) can be defined as a function satisfying the 

KG) = 


and the original boundary conditions when x =£ | and 

L(G) = oo when x = £ 

L(G) fife = 1 
Show that 

(aj) = AG(x, 


satisfies the original boundary-value problem. 

4.17. If the variables in polar coordinates are denoted by s and 6, the Laplacian 
operator V 2 is known to be 

a 2 l a l a 2 

V 2 = 1 1 

ds 2 s ds s 2 dd 2 

(a) Write the differential equation of motion for the vibration of a uniform 
thin circular membrane. 

{b) Let the membrane be held fixed at its outer rim, where s = s , and let the 
vibration be symmetrical with respect to the center. Show that the 
eigenvalue problem associated with such vibrations is described by zero- 
order Bessel's differential equation 

<Pr \_dr_ 
ds 2 s ds 

The solution is zero-order Bessel's function 

r(j) = J (Ps) 

(c) Show through the self-adjoint property of this eigenvalue problem that 

sJo(fos)Jo(PjS)ds =0 / #y 


where & and fa satisify 

Ufr ) = 

(d) By separation of variables, derive the eigenvalue problem for vibrations 
that are not symmetrical with respect to the center of the membrane. 

4.18. Show that the eigenvalue problem 

DV = /9 4 r 
r(0) = r'"(0) = r(L) = and r "(0) = 

has no solution. Can you reason out physically that there is no solution? 

4.19. Use the wave propagation concept to find the solution for Exercise 4.4 
for < / < L/c. How can you relate this function with the indicial response 
function ? 


4.20. A uniform bar of length 4a is fixed at x = and free at x = 4a. At 
/ = 0, the initial conditions for a longitudinal vibration are given by 

u(x, 0) = x < x < a 

u(x, 0) = 2a — x a < x <2a 
u(x, 0) = la < x < 4a 



(a) Show by concept of wave propagation that at t = 2a/c the displacement 
is given by 

u = —2" 

u = (x — 2a) a < x < 3a 

u = (4a — x) 2>a < x < 4a 

u = —\x < x < a 

(b) How would you find dujdt at t = 2a\c 

4.21. A semi-infinite beam has a hinged end at x = and extends to £ = -I- x . 
Show that (133) is the solution for the lateral vibration of this beam under the 
initial condition 

v (x, 0) = V{x) 


s ■ 

4.22. Find the approximate solution for the first mode of vibration of a beam 
with fixed ends : 

(a) By finite difference equations approximating the differential equation. 

(b) By finite difference equations approximating the integral equations. 

4.23. Furnish the missing steps in the derivation of (134). 

4.24. The boundary conditions for a uniform thin plate simply supported 
along its edge are 

w = and V 2 h> =0 on C 

Show that the eigenvalue problem for the vibration of this plate is the same as 
that for the vibration of a uniform membrane of the same contour. 

Hint: The boundary- value problem associated with (14) may be written as 

(V 2 - /5 2 )[(V 2 + £ 2 )r] = 
in which £ 4 = ph/ET. 


An Outline of Matrix 
Algebra in Linear 
Transformation of Vectors 

1. Introduction 

A matrix is an array of numbers taken together as a single mathematical 
entity. The dictionary meaning of the word matrix is "something in which 
other things are imbedded." In mathematics an array of numbers, which 
we call a matrix, also fits this description. Take for instance a set of linear 
simultaneous equations 

3^ + 4x 2 

2x 1 + lx 2 


= 1 
= 4 

= -3 


The two things that characterize this set of equations or distinguish it 
from other sets of linear equations are 

the coefficients of the unknowns 

and the constant terms 

Therefore, in these two arrays of numbers a set of linear simultaneous 
equations is imbedded. 





1 -3 

_2 7 


r *! 






If we wish to emphasize the fact that the unknowns are called x ± , x 2 , and 
x 3 and that the constant terms appear at the right-hand side of the equa- 
tions, we may write symbolically 




r n 







_x 3 _ 



And if we wish to represent a set of simultaneous equations in general, 
we may write symbolically 

A x = c (3) 

in which the symbol A stands for a square array of numbers (coefficients), 
x for a column of numbers (unknowns), and c for a column of numbers 
(known constants). 

Thus we have a shorthand notation of a set of linear simultaneous 
equations. This notation, called the matrix notation, achieves an economy 
of thought as well as an economy in writing. 

Notations alone, of course, do not make an algebra, which also deals 
with rules of operation. In (3) we have more or less indicated that there is 
a relationship denoted by the equal sign " = " and an operation denoted 
by the multiplication sign "•". The question now is what other kinds of 
operations can be defined and what algebraic properties do they possess. 
For instance, is there a matrix and an operation "— " such that the 

A • x — c = 

is equivalent to the one above? 
equivalent to (3) we may have 

Or, is there a matrix A -1 such that 

c = x 

Matrix algebra deals with questions of this nature. 

2. Vectors in n-Dimensional Space 

Although, historically, matrix algebra evolved from the study of linear 
equations, for our present exposition we shall use geometrical models to 
illustrate matrix operations. Such models have the advantage of revealing 
at once both the computational aspects and the algebraic properties of 
matrix operations. Geometrical models with more than three dimensions 
are, of course, difficult to visualize. Fortunately, methods in analytical 




geometry allow us to extrapolate, so that we shall not have to visualize 
more than three dimensions. 

It is well known that by a suitable choice of coordinate system we 
can represent any point in space by an ordered set of numbers, which we 
call the coordinates of the point. Also, by joining the origin of the co- 
ordinate system to a point in space, we define a free vector, and the set of 
numbers representing the coordinates of the point become the components 
of the vector. The three things, a point, a vector, and a set of numbers, 
therefore, are images of one another. We can thus use one kind of symbol, 
such as p, to denote any one of these things, or we can use a set of n 
numbers arranged in a column to denote the same things. 

P = 




Such a column of numbers is called a column matrix. 

3. Equality among Column Matrices 

Two vectors are the same if and only if all their respective components 
are equal. Two points coincide if and only if their respective coordinates 
are equal. Hence we can define the equality between two column matrices 
as follows. Let 

P = 




The equality relation 


or, in general 

-J>nA \An\ 

p = q 

Pi = qi P2 = a 2 m "Pn = q n 

Pt= a i / = 1 , 2, • • • , « 

An equality relation among column matrices is like the equality among 
numbers in that it is reflexive, symmetric, and transitive. 




4. Addition of Column Matrices 

(a) algebraic rules 

It is known in vector algebra that if p and q are two vectors there is an 
addition operation (parallelogram law) which yields another vector r. Or. 
symbolically, p + q = r (4) 

Two column matrices can also be added together to yield a third column 
matrix. Furthermore, vectorial addition is known to be commutative and 
associative. That is, 

p + q = q + p 

(p + q) + r = p + (q + r) 



We have shown that two column matrices can be added together and 
what the algebraic properties of the addition operation are. Now we may 
ask, "How is the sum to be determined?" The actual numbers that make 
up a column matrix, so that it represents a given vector, evidently depend 
on the coordinate system chosen. The same vector is represented by 
different columns of numbers in different coordinate systems. The question 
of how to obtain from given p and q the column of numbers making up 
r in (4) must also depend upon the choice of coordinate system. 

The arithmetic rules of matrix operations are based upon the use of a 
parallel coordinate system, in which the coordinate axes are all straight 
lines. The scaling of each axis is uniform, but it can vary from axis to axis. 
A three-dimensional parallel coordinate system is illustrated in Fig. 100. 
The vector p shown is represented by the column matrix 


A rectangular or Cartesian coordinate system is a special form of parallel 

coordinate system in which the coordinate axes are mutually orthogonal 

and the scaling is the same on all axes. 

Vectorial addition in a parallel coordinate system is particularly simple. 

It can be shown that the parallelogram law of addition is equivalent to the 

following rule. If 

Pi H\ 

p* q-. 






p+ q 

'(pi + ?i) 

(P2 + ?2> 



ipn + q n )_ 

In other words, each component of the vectorial sum is the sum of the 
respective components of the vectors making up the sum. 

Figure 100 

Let us remark that although one can deduce the algebraic property of 
addition, (5) and (6), from the arithmetic rule (7) the former has an 
independent existence of its own. 

5. Multiplication by a Scalar and Linear Dependence 


we can write 

P+ P 

2p = q or p = iq 

Thus a column matrix can be multiplied by a scalar, and the equality 

q = ?P (8) 

implies that in a parallel coordinate system 

q i = 'kp i for all i = 1 , 2, • • • , n 


Geometrically, (8) implies that q and p have the same direction but differ 
in length. 

A vector r is said to be a linear combination of p and q if 

r = ap + /5q 

in which a and ($ are scalar quantities. Geometrically this means that the 
line segments representing the vectors p, q, and r are coplanar. The 
vectors p, q, and r are in the meantime said to be linearly dependent. In 
general, a set of vectors p (1) , p (2) , . . . , p {i) is said to be linearly dependent 
if for a certain set of scalar quantities, a l5 a 2 , . . . , a i? which are not all 
zero, the following equation holds: 

o^pW + <x 2 p< 2 >, • • • , <x,p<*> = (9) 

The proof of the following lemmas is left to the reader as exercises. 

Lemma 1. Any one of a set of linearly dependent vectors can be ex- 
pressed as a linear combination of the others. 

Lemma 2. If p, q, and r are linearly dependent and r, s, and t are 
linearly dependent, then p, q, s, and t are linearly dependent. 

6. Linear Transformation of Vectors — Algebraic Rules 

A set of n functions of n variables 

Vl ~ r\\ x ii x 2> ' ' ' J x n) 
2/2 == ^ 2\ X \'> X 2-> ' ' ' 1 X n) 

Vn — Fn\ X l-> X 2> ' •> X ) 

can be considered as a functional relationship between two vectors 

y = f(x) (io) 

since, given a column matrix x, one can find a corresponding column 
matrix y, and y may be considered as a vectorial function of a vector x. 
There are many physical and mathematical models for the relationship 
(10). We cite three such models for illustration. 

(i) x is the location of a generic material point in a deformable body 
before deformation, and y is the location of the same material point after 
the body has deformed. In this case (10) describes how the body is 

(ii) x is the location of a generic spatial point in a steady-state fluid flow. 
y is the velocity of the fluid particle occupying the point x. In this case 
(10) describes the velocity distribution of the flow. 


(iii) x is a column of numbers representing the coordinates of a fixed 
point, according to a certain coordinate system, y is another column of 
numbers representing the coordinates of the same point, according to 
another coordinate system. In this case (10) describes a transformation of 

In comparing (i) with (iii), we see that in one case the coordinate system 
is fixed, whereas the point moves, and in the other case the point is fixed, 
whereas the coordinate system changes. From a mathematical point of 
view the difference in the situations hardly matters. We can, therefore, 
always speak of (10) as representing a transformation of x into y. 

The simplest example of a transformation is that represented by (8), 
in which the vector q is obtained by "lengthening" p without a change in 
direction. Or we can consider (8) as representing a uniform change of 
scale factor of the coordinate system. Let us now conceive another class 
of transformations, which is rather simple and yet of great utility in 
analyzing physical and mathematical problem. These transformations are 
called linear and are denoted by bold sans serif letters such as A, B, L, and 
M. 1 A linear transformation is supposed to have the following properties: 

(I) Every point in space has a transformation and the origin remains 
fixed. Or, A(x), B(x), etc., are uniquely defined for every x and A(0) = 
0, B(0) = 0, etc. 

(II) Points lying on a straight line remain colinear. 

Without going into the method of carrying out this transformation, we 
can deduce the following lemmas successively from the properties postu- 

Lemma 1. Coplanar points remain coplanar because, unless a plane 
remains a plane, some straight lines on the plane will become curved. 

Lemma 2. Two intersecting lines remain intersecting, and two coplanar 
parallel lines remain coplanar parallel. This follows if we take (i) to mean 
that oo remains at oo and no finite point goes to infinity. 

Lemma 3. A parallelogram remains a parallelogram. 

Lemma 4. Linear transformation is distributive; that is 

A(p + q) = A(p) + A(q) (11) 

Let p + q = r 

1 For the moment these symbols have not yet been tied in with the similar symbol in 


Then, before transformation, p and q are two adjacent sides of a parallelo- 
gram and r is the diagonal in between. After transformation, the paral- 
lelogram is still a parallelogram, with A(p) and A(q) forming its two 
adjacent sides and A(r) forming the diagonal in between. Hence 

A(p) + A(q) = A(r) = A(p + q) 

Lemma 5. Linear transformation preserves all linear dependence 
First we see that 

A(2p) = A(p + p) = A(p) + A(p) = 2A(p) 

and, by similar reasoning, 

A(Ap) = AA(p) (12) 

Hence, if 

r = ap + /5q 

A(r) = aA(p) + /?A(q) (13) 

Lemma 6. A linear transformation followed by a linear transformation 
is still a linear transformation. This is obvious from the postulated 
properties of linear transformation. Symbolically, if 

A(p) = q and B(q) = r 

r is related to p by a linear transformation 

r = C(p) = B[A(p)] 

C = (BA) (14) 

This is equivalent to a multiplication operation. We say by (14) that C is 
obtained by premultiplying A by B. 

Lemma 7. It follows from the foregoing that a succession of linear 
transformations is still a linear transformation. Symbolically, if C = B A 
and E = D C, then 

E = [D(BA)] (15) 

Furthermore, the multiplication operation is associative or 

[D(B A)] = [(D B)A] 

because, by the definition of multiplication operation, 

[D(B A)](p) = D[(B A)(p)] = D[B{A(p)}] 

[(D B)A](p) = [D B] {A(p)} = D[B{A(p)}] 

Thus we can write (15) simply as 

E = D B A 




Lemma 8. The sum of the results from two separate linear trans- 
formations on the same vector is also a linear transformation. This is to 
say, if 

A(p)+B(p) = u (16) 

there is a linear transformation C, such that 

C(p) = u 

In other words, the relation between p and u, as defined by (16), is also a 
linear transformation. To show this, we observe first that (16) satisfies 
the property (I). All is left now is to show that (16) also satisfies property 
(II). This can be done as follows: 

The necessary and sufficient condition for three points p, q, and r to lie 
on a straight line is that (see Fig. 101) there is a scalar parameter a so that 

ap + (1 — a)q = r 

Since A and B are linear transformations, according to (13), 

aA(p) + (1 - a)A(q) = A(r) 

aB(p) + (1 - a)B(q) = B(r) 
Now if 

A(p)+ B(p) = u 
A(q) + B(q) = v 
A(r) + B(r) = w 

then by adding the equations in (17) and utilizing the foregoing equations, 
we have 

au + (1 — a)v = w 





Hence the transformation (16) preserves the colinearity and is a linear 
transformation representable by a symbol C. This completes the proof of 
this lemma. 
If we define 

(A + B)p = A(p) + B(p) (18) 

we may write 

(A + B) = C (19) 

Thus there is an addition operation among linear transformations. 
Furthermore, from (5), (6) and (8) the addition operation is commutative 
and associative. 

A + B = B + A 

A + (B + C) = (A + B) + C 

Note that the operation (14) on the other hand is not commutative. 
That is, in general, 

A B(p) ^ B A(p) 
or (21) 


7. Linear Transformation in Parallel Coordinate Systems 

In Art. 6 we made no mention of the method by which linear transforma- 
tion can be carried out. The algebraic rules were deduced strictly from 
the geometrical properties of the transformation. To determine how a 
linear transformation can be carried out, we refer again to a parallel co- 
ordinate system. 





where the elements in the column matrices represent the coordinates of x 
and y in a parallel coordinate system. If x and y are related by a linear 

y = A(x) 

then y l3 y 2 , . .-'. . ,y n are linear combinations of x v x 2 , 
Vi = ^li^i + #12^2' ■ * , o, ln x n 

Vi = fl 21^1 "T~ ^22^2' ' a 2n X n 







It is not possible to show here that a linear transformation in Euclidean 
geometry must take the form of (23). However, one can easily verify that 
(23) represents a transformation that satisfies all the requirements of being 
a linear transformation. 

The linear transformation (22) therefore embodies a set of linear 
simultaneous equations relating the coordinates before to those after 
transformation. These equations are imbedded in the matrix equation 

y = A(x) = A • x = A x 



a n a 12 • • • a 

a 9 ,a 

21" 2 2 


• a 

The rule of premultiplying a column matrix by a square matrix is given by 

(23), or n 

Vi = 2 a a x j / = 1, 2, • • • , w (23a) 


where a {j is the element in the ith row and yth column of the square 
matrix A. We can now deduce the following rules: 

(i) If, for all x, 
we write 

A(x) = B(x) 

A = B 

This equality implies that the transformations A and B yield identical 

2 a a x i = 2 b a x o i = \,2,- " ,n 

j=i j=\ 

for all x t . In order for this to be true, we must have 

i = 1,2, • • • ,n 

j = \,2,---,n 

In other words, two n x n square matrices are said to be equal if and 
only if all the corresponding elements are equal, 
(ii) If, for all x, according to Lemma 8, 

A(x) + B(x) = (A + B)x = C(x) 
we write 

A+ B = C 


This implies 

n n 

2 (an + b ij )x j = J c ij x j i = 1 , 2, • • • , n 

3=1 3=1 


i = 1,2, •*-,« 

«w + 0« = c u (24) 

/= 1,2, •••,« 

In other words, the sum of two « x n square matrices is also an n x n 
square matrix, whose elements are obtained by adding the corresponding 
elements of the original two matrices, 
(iii) If for all x, according to Lemma 6, 

B[A(x)] = C(x) 
B A = C 

n n n 

2 2 b ik a kj x j = J c {j xj i = 1, 2, 

j=lk=l j=l 


i= 1,2,- 

• • , n 

2* ^ik a ki = C ij 

■ = l 


• - , n 

we write 
This implies 


1=1.2. ",n 


In other words, when two n x n square matrices form a product, which is 
also an n x n matrix, the element at the /th row and y'th column of the 
product is determined by the entire /'th row of the first matrix and the 
entire y'th column of the second matrix, according to (25). 

8. Summation Convention and a Summary 

We note the following in all foregoing equations that describe the 
relationships among elements of matrices in matrix operations: 

(i) All subscript indices that are to be summed from 1 to n appear 
precisely twice in a term. 

(ii) All subscript indices that are not summed but are to take on any 
value from 1 to n appear precisely once in each term of the equation. 
Since this is always the case, we can simplify our notation by deleting all 
summation signs and omitting notations such as / = 1, 2, . . . , n. 

The convention then adopted is as follows: 

(i) If an index appears twice in a given term, a summation process is 
understood to exist without the explicit use of a summation sign. The 
index is called a dummy index. 




(ii) If an index appears only once in each term of an equation (except 
the term zero), it is understood that this index may take on any value from 
1 to n. The index is called a. free index. 

An equation such as (25) can be written without ambiguity as 

£>ik a ki = c a 

in which k is a dummy and / andy are free. 

With this convention, we can summarize the result of the previous 
articles by the following table: 

Matrix Relation* 

Relation Among Elements 

x = y 

x i = Vi 

x + y = z 
Ax = y 

x i + Vi = z i 
toi = Vi 

A x = y 

A = B 

A + B =C 

AA = B 

a ij x j == Vi 
Gil + bij = Cij 

Xa^ = bn 

AB = C 
A BC = D 

a ijbjk — c ik 
QijDjkCkm = Cjm 

* We have omitted parenthesis and dots in matrix products. Greek letters represent 
scalar quantities. 

The operations among matrices shown have all the properties of 
arithmetic operation among numbers, except that multiplications are not 
commutative. Thus we can deduce relations as follows: 


(A + B)(x + y) 

(A + B)(A .+ B) 

A 2 = A A 

Ax+Bx + Ay+By 
A2 + AB+BA+B 2 




Also, if the elements of the matrices are functions of a scalar variable such 
as time t, we may perform differentiation and integration processes with 
respect to t, since they are the limits of processes involving only additions 
and scalar multiplications. Hence 





d a d „ 
dt dt 


B + A-B 



Note that 





9. Inverse Operation 

The linear transformation 

A x = y 

is supposed to be unique only in the sense that for every x there is a 
unique y. It is, however, not necessary that a unique x be transformed 
into every y. In other words, although every point is transformed into 
only one point, two different points may be transformed into the same 
point. If however, a one-to-one correspondence exists in the transformation, 
then the transformation has an inverse. 

A x = y (26) 

If we are given y, x is to be found by solving the set of simultaneous 
equations (23). Now, it is known that (23) will have a unique solution if 
and only if the determinant 

a u a 12 • • - a ln 

a<„a 99 - - • a< 

A = 

a nl a n2 

• • • a. 

= W J ^ 

Thus the condition for the existence of an inverse to A is that |A| ^ 0. 
We say that A is nonsingular if it has an inverse. Let the inverse of A be 
denoted by the symbol A -1 . The inverse transformation of (26) is then 

A -1 y = x 

Evidently, if A -1 exists, it must also be a linear transformation. 
The square matrix 

"l ••• 0" 
1 ••• 

1 = 



is called an identity or unity matrix because it represents the identity 

I x = x 

where every point remains fixed. 
From (26) and (27) we see that 

A A -1 y = Ax = y = Iy 

A -1 A x = A -1 y = x = I x 


A A 1 = A 1 A = I 

a ik a kj == l ij == °ij 


where d {j is called Kronecker's delta 2 defined by 

d tj = 1 when i = j 
d tj = when i ^ j 

In ^-dimensional space (28) represent n sets of n simultaneous equations. 
A typical set is 

a 21 a 12 ~ 1 + a 22 a 22 ~ 1 + • • • a 2n a n2 ~ x = 1 

«31«12 _1 + «32«22 _1 + • ' ' #3 A2 -1 = 

a m a i2 1 + ^2^22 x + ' ' ' a nn a n2 X = 

If elements a tj are known, the elements a _1 of the inverse can be found by 
solving these simultaneous equations. According to the so-called Cramer's 
rule, the result is 

= cofactorof^inK-l 



N = |A| # o 

Although (29) is a concise definition for the elements of the inverse 
of a nonsingular square matrix, the formula is not used for numerical 
computation, except when the matrix has very few elements, such as a 
3x3 matrix. This is because there are other schemes of numerical 
computation that are less laborious. For our purpose it is sufficient to 
know that a nonsingular square matrix has an inverse whose elements are 
uniquely determined. 

2 There is no reason to change i into d in (28), except that Kronecker's delta is an 
established symbol. 


From the definition of inverse matrix and identity matrix we have the 
following equations: 

C = AB 
A 1 C = A 1 A B = I B = B 
B 1 A 1 C = B 1 B = I 
B 1 A 1 C C 1 = I C 1 = C 1 


B -i A -i = c -i 

In other words 

(A B)-i = B-i A" 1 (30) 


(A B C)- 1 = C- 1 (A B)- 1 = C- 1 B- 1 A- 1 (31) 

10. Scalar Products and Transposition 

So far, the matrix products that have been defined are A x and A B. 
Products such as x y and x A have yet no meaning. In this article we 
define the meaning of some new products. 

(a) scalar product in cartesian coordinates 

In vector analysis two vectors can form a scalar product with certain 
physical or geometrical significance. 

Let the following notations be adopted : 

|x| and |y| = lengths of two vectors represented by x and y 
p(x, y) = scalar product of the two vectors 
6 = the angle between the two vectors 
The geometrical meaning of scalar product is then 

p(x, y) = |x| |y| cos 6 (32) 

If the elements of the matrices x and y stand for the components of the 
vectors in a Cartesian coordinate system with scale factor equal to unity. 
it is known that 

p(x, y) = x lVl + x 2 y 2 + ,••', x n y n = x { y t 
In matrix notation we write 

p(x, y) = [x x x 2 --x n ] 

y n . 

= xy 



The components of the first vector are now arranged as a row, instead 

of a column, to conform to the pattern of element-pairing procedure 

previously established in defining products A x and A B. In these two 

cases we take a row of the first matrix and pair its elements with a column 

of the second matrix. The sum of the products of the pairs becomes an 

element of the product. The symbol x denotes a row matrix that is 

obtained by making the elements in the column matrix x into a row. 


J x y = y x 

(b) scalar product in general parallel coordinates 
If the elements of the matrices x and y are the components of two 
vectors referring to a general parallel coordinate system other than a 
rectangular Cartesian system, the matrix product x y or y x is no longer 
the scalar product defined by (32). To obtain p(x, y) in a general parallel 
coordinate system, we may first transform the given coordinate system 
into a Cartesian coordinate system. It must be realized that the scalar 
product p(x, y), which is a scalar quantity of definite geometrical meaning, 
remains invariant under any transformation of coordinates. In other words, 
it is a quantity of definite numerical value, no matter what kind of reference 
system is used for computation. 

Let x and y be two column matrices whose elements represent the 
components of two vectors in some given parallel coordinate system. Let 
A be a square matrix representing a transformation that transforms the 
given coordinate system into a Cartesian coordinate system with unit 
scale factor. We use the article "a" in the foregoing sentence, instead of 
the article "the," because the transformation is not unique, for two 
coordinate systems may both be Cartesian but with differently oriented 
axes. The matrices u and v defined by 

u = A x and v = A y (34) 

thus represent the components of the same two vectors referred to a 
Cartesian coordinate system. Because of the invariance of scalar product 

pi*, y) = />(u, v) = U v 
the symbol u stands for a row matrix obtained by rearranging the column 
matrix u into a row. To conform to the established pattern 3 of matrix 
multiplication, we may write 

u = [u ± u 2 - ■ • U n ] = [XjX 2 - • -x n ] 

= x A 

#12^22 ' * a n2 

#i«#9« * * * a nr , 

That is, by pairing the rows of the first matrix with the columns of the second matrix. 


In longhand this equation is the same as the first one of (34), in which the 
symbol A stands for a matrix that is obtained by rearranging the elements 
of the matrix A in such a way that the rows of A become the columns of A 
and the columns of A become the rows of A without otherwise disturbing 
the order of things. In other words, the elements in A and A are related by 

an = <*ji or a a = a H 

1 11U5 

p(x, y) = p(u, v) = uv = xAAy 


AA= M 


p(x, y) = x M y 



(i) If A = B,then 

B = A or 
(ii) If A B = C, 


B A = C or 



(Ax) = 

x A 

The operation of converting the columns of a matrix into a row, and 
vice versa, is called a transposition. The result of the transposition is 
called the transpose and is denoted by a bar over the letter representing 
the matrix transposed. Thus A is called the transpose of A and x is 
the transpose of x. The following simple formulas pertaining to trans- 
position can easily be proved. 

B A (37) 

(A B C) = C B A (38) 


If |x| represents the length of the vector x, according to (32), 

|x| 2 = p(x, x) = x M x (39) 

The matrix M is consequently called the metric matrix of the coordinate 
system. For a given parallel coordinate system M is uniquely determined, 
even though the matrix A in (34) is not. To prove this statement, let 
us assume that there is another matrix, say N, that will yield the length 
of a vector through (39); then 

x N x = x M x 


n^xj = m^x^ 


Since x is arbitrary, the two sides can be equal only if the coefficients are 

"ll = ™11 («12 + «2l) = ( m 12 + W 2l) "22 = ™22> ' ' • 

or, in general, 

(*« + «n) = (w« + ?%) 
N + N = M + M () 

But, according to (35) and (37), a metric matrix is always symmetrical in 
the sense that M = M because 

M = AA M = (AA) = AA=M 

So must __ 

N = N 
Thus (40) reduces to 

N = M 

(e) orthogonality relation between two vectors 

Two vectors are said to be orthogonal to each other if their scalar product 
is zero. Geometrically, we see that 

P (x y) = |x| |y| cos 6 = 

implies that 6 = ±7t/2. For two vectors x and y to be orthogonal in a 
general parallel coordinate system the condition is 


In a Cartesian coordinate system, 

M = 1 
the orthogonality relation is expressed simply as 

xly = xy = 


Absorbers, dynamic vibration, 138-146 
Accelerometers, 76, 77-78 
Admissible functions, 275-277, 282-286 
Amplitude, 5 

complex, 6, 8, 14 
Amplitude modulation, 21 
Amplitude ratio, 102, 111 

see also Modal vectors 
Analog computer, 227 
Analogs, electrical, 221-227 
Analogous systems, 221 
Analyzer, wave, 29 
Aperiodic motion, 13, 115 
Argand's diagram, 7, 8, 16, 45, 158, 159 

Baron, M. L., 315 

Boundary conditions, 238, 240 

imposed or geometrical, 241, 248, 285, 

linear homogeneous, 241, 251, 264, 270 
natural or dynamical, 241, 248, 286, 
Boundary-value problem, definition of, 

Bromwich integral, 51 

Cauchy's principal value, 49 
Centrifugal force, excitation by, 125, 149 
Chain systems, 212 

Characteristic equation, 114, 116, 176, 
see also Frequency equation 
Characteristic values, see Eigenvalues 
Circular frequency, 5 

Collatz, L., 209, 286 
Column matrix, 325-328 
Columns, buckling of, 302 

vibration of, 300-302 
Compensation of instrument response, 

Complex amplitude, 6, 8, 14 
Complex damping, see Complex stiffness 
Complex frequency, 14 
Complex number representation, 6, 24- 

of damped oscillations, 14, 27, 88 

of harmonic oscillations, 6, 24-26, 85 
Complex stiffness, 86, 88 
Compound pendulum, 56 
Concentrated mass on elastic bars, treat- 
ment of, 265-268, 269 
Constraints, 9, 169, 170, 238 

effect on natural frequencies, 292-294 
Continuous frequency spectrum, 304, 305 
Coulomb damping, systems with, 62-68 
Couplings, inertia and elastic, 106 

effect on natural frequencies, 136-138 
Cramer's rule, 337 
Crandall, S. H., 89 
Critical damping, 14 

resistance of galvanometer, 69 
Critical speeds of shafts, 123-128, 146- 

of the second order, 155 
Curreri, J. R., 135 

D'Alembert's principle, 224 
Damped oscillation, 12-14, 115-116 




Damper, linear or viscous, 12, 114, 188 
Damping, 13 

Coulomb, 62 

linear or viscous, 12 

structural or hysteresis, 82-89 
Damping constant, 13 
Damping of accelerometers, 78 
Damping of galvanometers, 68, 69, 70-71 
Damping of vehicle suspension, 81 
Damping ratio, 16 
D'Arsonval galvanometer, 68 

see also Galvanometer 
Datum node, 224 
Decrement, logarithmic, 15 
Degrees of freedom, definition of, 169 
Den Hartog, J. P., 65, 140, 153 
Diagonalization of symmetrical matrices, 

Difference equation, see Finite difference 

Dirac function, 34, 267 
Dirchlet's condition, 28 
Dissipation, energy, 31, 188, 228 
Divergence theorem, 270 
Draper, C. S., 73 
Duhamel's integral, see Superposition 

Dummy index, 173 
Dynamical boundary conditions, see 

Boundary conditions, natural 
Dynamic state, 11 
Dynamic vibration absorber, 138-146 

Edgar, R. P., 320 
Eigenfunctions, 252 

infinite series expansion by, 254, 265- 

orthogonality of, 260-274 
Rayleigh's quotient of, 277 
Eigenvalues, 179 

Determination by matrix iteration, 

maximum-minimum characterization 

of, 278-279 
minimum characterization of, 185, 278 
negative, 302 

see also Eigenvalue problem 
Eigenvalue problem, of a differential 
equation, 249-260 
of an integral equation, 271-274 

Eigenvalue problem, of a matrix equa- 
tion, 179 
of a symmetrical matrix, 209 

Eigenvectors, 179 

see also Modal vectors 

Elastically supported ends, 264, 269 

Elastic body, configuration of, 237-238 

Elastic constants, 106, 171, 172, 176 

Elastic coupling, 106, 137 

Elastic energy, see Potential energy ex- 

Elastic matrix, 175, 208 

Electrical analogy, 221-227 

Electro-mechanical systems, 227-231 

Enclosure theorem, 208-209 

End conditions, 241, 243, 247, 257-258, 
see also Boundary conditions 

End mass, treatment of, 265-267, 269 

Energy dissipation, 30, 65, 83, 157 

Energy methods, see Rayleigh's method 
and Rayleigh-Ritz method 

Energy relation, 10, 31, 186, 228, 229, 

Equalization, of instrument response, 

Error of finite difference approximation, 
of galvanometer response, 72 

Essential boundary conditions, see 
Boundary conditions, imposed 

Euler critical load, 302 

Excitation, 40, 47, 48, 116 
see also Signal 

Expansions, infinite series, 238, 254, 265- 
267, 275, 288, 297 

Finite difference approximations, 312— 

for differential equations, 312-315 
for integral equations, 315-316 
Forced vibrations, of elastic bodies, 296- 

of several-degree-freedom systems, 

185-188, 189-192 
of single-degree-freedom systems, 17- 

19, 23, 28-29, 31-39, 62, 65-68, 83- 

of systems with Coulomb damping, 




Forced vibrations, of systems with struc- 
tural damping, 83-86 
of two-degree-freedom systems, 116— 
Foss, K. A., 192 

Fourier series, 28, 29, 238, 255, 260 
Fourier transform, 48, 309, 311* 
Freedom, degrees of, 169 
Free free- vibration, 195 
Free index, 173, 335 

Free vibrations, of elastic bodies, 236- 
of several-degree-freedom systems, 

175-178, 188-189 
of single-degree-freedom systems, 3-17 
of systems with Coulomb damping, 

of systems with structural damping, 

of two-degree-freedom systems, 100- 
Frequency, 5, 10 
beat, 22 
circular, 5 
complex, 14, 115 
damped, 14 
imaginary, 302 

natural, 5, 102, 107, 112, 176, 292 
Frequency equation, of bars in lateral 
vibrations, 256, 257, 258 
of bars in longitudinal vibration, 253 
of chain systems, 213 
of infinite-degree-freedom systems, ap- 
proximate, 281 
of nonrotating shaft, 148 
of rectangular membrane, 259 
of rotating shafts, 151, 152 
of several-degree-freedom systems, 176 
of two-degree- freedom systems, 102, 
106, 111 
Frequency modulation, 21 
Frequency ratio, 19 
Frequency response, 23, 41-43 
of galvanometer, 70-71 
relation with indicial response, 46-51 
see also Steady-state response and 
Transfer function 
Frequency spectrum, continuous, 304, 
discrete, 304 

Friction, dry, 62 
Functional, 275 

Gain, 42 

Galvanometer, equation of operation, 
69, 230 
frequency response of, 70, 71 
optimum damping of, 71, 73 

Generalized coordinates, 169-173, 238 

Generalized forces, 185-186, 300 

Generating functions, set of, 280 

Graeffe's method, 116 

Gravity effect on critical speed, 153-155 

Gravity pendulums, 55-56 

Green's function, 272, 273, 317, 319, 320 

Green's theorem, 270 

Gyroscopic effect on critical speed, 146- 

"Half" degree of freedom, 122 
Hallowell, F. C, 89 
Harmonic analyzer, 29 
Holonomic systems, 170 
Holzer's method, 212-220 
Hysteresis damping, see Damping, struc- 
Hysteresis whirling, 156-160 

Identity matrix, 203, 337 

Imaginary frequency, 302 

Imaginary pivot points, 313 

Imposed boundary conditions, 241, 248, 

285, 292 
Impulse response, defined, 36 
of an elastic body, 303 
of several-degree-freedom systems, 

187, 191, 192 
of single-degree-freedom systems, 36, 

37, 38 
relationship with frequency response, 
48, 49, 50, 51 
Impulse response matrix, 191, 192 
Incident wave, 308 

Indicial admittance, see Indicial response 
Indicial response, defined, 32 
of an elastic body, 303 
of several-degree-freedom systems, 

191, 192 
of single-degree-freedom systems, 36, 
37, 38 



Indicial response, relationship with fre- 
quency response, 48, 49, 50, 51 

Indicial response matrix, 191, 192 

Inertia constants, 106, 172 

Inertia coupling, 106, 107, 137 

Inertia matrix, 175 

Infinite elastic body, 304-311 

Infinite degrees of freedom, 238, 281 

Infinite series formulation, 287-290 

Influence coefficients, 108-110, 208, 271, 

Initial conditions, 5, 12, 13, 14, 28, 32, 
35, 103, 115, 116, 178, 189, 191, 240, 
254, 306, 311 

Input, see Signal 

Instruments, seismic, see Seismic instru- 

Integrable constraints, 169 

Integral equation formulation, 271-274 

Interior pivot points, 313 

Inverse Laplace transform, 51 

Inverse Nyquist locus, 46 

Inversion of matrix, 336-338 

Iteration, matrix, 197-202 

Kinetic energy, 10, 106, 170, 284 
Kinetic energy expressions, for elastic 
body, 284, 287, 289, 290 
for several -degree-freedom systems, 

171, 172 
for single-degree-freedom systems, 11 
for two-degree-freedom systems, 106 
Kimball, A. I., 83, 89, 156 
Kirchhoff's law, 221, 223, 224, 230 
Kronecker's delta, 287, 337 

Lagrange's equation, 106 

Lagrangian description of deformation, 

Laplace transform, 44, 51 

Laplacian operator, 270 

Lateral vibration of bars, 245-248, 255- 
258, 309-311 

Lazan, B. J., 83 

Lees, S., 73 

Leonhard, A., 50 

Lin, C. C, 116 

Linearization of systems in small oscilla- 
tions, 55-58, 171-173 

Linear systems, definition, 31, 171 

Linear transformation of vectors, 179, 

328, 334 
Locii, transfer, 45-46 
Logarithmic decrement, 15 
Logarithmic spiral, 16 
Longitudinal vibrations of bars, 240-243, 

252-255, 305-309 
Longitudinal waves, 305-309 
Loop current, 221, 222, 224 
Lovell, D. E., 83, 89, 156 
Lowest natural frequency, determination 

of, 200-202 

MacDuff, J. N M 135 

Magnification factor, 19, 23, 42 

Matrix iteration, 197-202 

Matrix notation, 174 

Maximum-minimum characterization of 
eigenvalues, 278-279 

Maxwell's reciprocal theorem, 110, 147, 

Membrane, vibrations of, 248, 258-260 

Method of finite difference, see Finite 
difference approximations 

Metric matrix, 340 

Minimum characterization of eigen- 
values, 278 

Modal matrix, 178 

Modal vectors, 177 

Modes of vibration, see Principal modes 

Mohr's circle, 107 

Moving coil instruments, 68 

Moving external force, 299 

Myldestad, N. O., 89 

Natural frequency of vibrations, of elas- 
tic bars and beams, 253, 256, 257, 258 
of light beams and shafts carrying 
masses and disks, lateral, 111, 134, 
of several-degree-freedom systems, 175 
of shafts, torsional, 212-220 
of single-degree-freedom systems, 4 
of stretched membrane, rectangular, 

of two-degree-freedom systems, 107 
Natural modes, see Principal modes 
Negative spring, 58 

Network analogy to vibration systems, 



Neumark, S., 89 

Nonholonomic systems, example of, 170 
Nonintegrable constraints, 170 
Nonlinear systems, approximate analy- 
sis, 55-62 
Nonsingular matrix, 336 
Normal coordinates, 181 
Normalized modal vectors, 177, 210 
Norton, A. E., 160 
Nyquist diagram, 45 
Nyquist locus, 46 

Oil whip, 160-161 

Optimum damping of accelerometers, 78 

of dynamic absorbers, 145-146 

of galvanometers, 70-73 
Optimum design of dynamic absorbers, 

Orthogonality of eigenfunctions, 260-272 

of modal vectors, 108, 181-182, 197, 210 
Orthogonality relation, defined, 260, 261, 

Orthogonalization, 183 
Over-all transfer locus, 46 
Overdamped systems, 13 

Parallel coordinate systems, 326 

Partial differential equation of motion of 

elastic bodies, 239 
Particular integral, 39 
Pendulum, compound, 56 

simple, 55 

spring-loaded, 56 

torsional, 52 
Period, 5 

Periodic forces, 28 
Phase difference, 6, 23 
Phase plane, 11 
Phase-shift distortion, 78 
Phase trajectory, 11 
Phase velocity, 11 
Pian, T. H. H., 89 
Pickups, vibration, 74-78 
Piecewise-linear systems, 59-68 
Pivot points, 312 

Plates, thin, vibration of, 249, 271, 322 
Positive definite systems, definition of, 

Potential energy expression, 10 

of beams carrying weights, 134 

Potential energy expression, of elastic 
bodies, 93, 287, 290 
of single-degree-freedom systems, 10 
of semidefinite systems, 192, 193 
of several -degree-freedom systems, 

171, 172, 174 
of two-degree-freedom system, 105, 
Potential forces, 185 

Principal coordinates, 104, 118, 180, 181 
Principal modes, 102, 104, 175 
Principle of superposition, see Superposi- 
tion principle 
Propagation of waves, 305-308 

Quality factor, 15 

Rayleigh's dissipation function, 188 
Rayleigh's method, 10, 132-135, 185 
Rayleigh's quotient, defined, 111, 184, 

as function of amplitude ratio, 111 

of admissible functions, 275 

of essentially admissible functions, 284 

of infinite series, 276 

of vectors, 111, 184 

stationary values of, 112, 132, 185, 
Rayleigh-Ritz method, 279-292 
Reflexion of waves, 308 
Reluctance pickups, 75 
Repeated roots of frequency equation, 

123, 196-197, 260 
Resonance, 19 
Response time, 72 
Robertson, D., 160 
Rotating shaft, vibrations of, 123-128, 

Rotating vector, 7-9 
Runge, C, 23 

Salvadori, M. G., 315 
Scalar product of vectors, 338-341 
Scarborough, J. B., 29 
Seismic instruments, 73-79 
Seismograph, 73-74 
Self-excited vibrations, 82 
Semidefinite system, 192-196 
Semi-infinite elastic body, 304-311 
Separation of variables, 249-252, 259 



Signal, 40 
Soholnikoff, I., 46 
Space variable, 249, 250 
Spectrum, frequency, 304-305 
Stability of elastic systems, 302 
Stable equilibrium, 171, 192, 302 
Standing waves, 308-309 
Static deflection, defined, 19 
Stationary properties of eigenvalues, 

112-113, 184-185, 277-278 
Stationary values of Rayleigh's quotient, 

112, 184, 277-278 
Steady-state response, 18, 39 

of single-degree-freedom systems, 17- 

24, 28-29 
of several-degree-freedom systems, 

of two-degree-freedom systems, 117— 

with Coulomb damping, 65-68 
with structural damping, 83-85 
Step function, unit, 32 
Stieljes integral, 34 

Stiffness numbers, see Elastic constants 
String, vibration of, 244-245 
Successive reduction of eigenvalue prob- 
lems, 206-207 
Summation convention, 173, 334 
Superposition principle, 31-36 
applied to elastic body, 303 
applied to several-degree-freedom sys- 
tems, 187-188, 190-191 
Suspension of vehicle, 79-81 
Sutherland, R. L., 22 
Symmetrization of general eigenvalue 
problems, 209-212 

Tensor notation, 173 

Time-invariant systems, 33 
Time-phase angle, 5 
Time-variable, separation of, 249-252 
Timoshenko, S., 140 
Torsional vibrations, 52, 212-220, 243 
Transducers, seismic, see Seismic instru- 
Transfer function, 43, 44 
Transfer locus, 43, 46 
Transmissibility, 42 
Transient response, 18, 39-41 

see also Indicial response 
Triangular matrix, 211 

Unit impulse, 34 

Unit step function, 32, 46 

Unity matrix, 337 

U-tube, oscillation of liquid in, 52-54 

Vectorial addition rules, 326-327 
Vectorial diagram of steady-state solu- 
tion, 27 
Vehicle suspension, 79-81 
Velocity of longitudinal waves, 306 
Velocity pickups, 76, 77 
Velocity sensitive, 76 
Veubeke, B. M. F., de, 200 
Vibration absorbers, 138-146 
Vibration controls, 135-138 
Vibrometer, 76 

Wave equation, 248 

Wave phenomena, 304-311 

Work done by damping forces, 30, 65, 

Work done in terms of generalized co- 
ordinates, 186 

Work done per cycle, 30-31 










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